With the paradox of fiction, it is not particularly easy to find a case. Obviously people in general do not affirm all three claims; there are many people who definitely affirm one or two, but it's harder to find someone who is actually committed to something like the paradox itself. And, of course, the claims in question would naturally tend to be scattered even in author who was committed to the paradox, since no one would just baldly affirm all three members of what seems like an inconsistency without some qualification. But a good candidate for someone who is committed to all three claims in the paradox of fiction is David Hume, and so it makes sense to look at Hume's claims on fiction in order to see, first, what makes it plausible that he could be committed to all the claims in the paradox and, second, whether there are any qualifications or nuances that give him a way out.
(1) It is irrational to be moved to feel for what we know does not exist.
One might think that Hume avoids this, because Hume holds that for the most part the passions can't be either rational or irrational. But in fact he makes two exceptions:
...passions can be contrary to reason only so far as they are accompanyed with some judgment or opinion. According to this principle, which is so obvious and natural, it is only in two senses, that any affection can be called unreasonable. First, When a passion, such as hope or fear, grief or joy, despair or security, is founded on the supposition or the existence of objects, which really do not exist. Secondly, When in exerting any passion in action, we chuse means insufficient for the designed end, and deceive ourselves in our judgment of causes and effects. (T 188.8.131.52)
We can set aside means-end irrationality as not relevant to our concern. But we do clearly have a statement that passions can be called unreasonable when they are accompanied by a judgment that something exists when it really does not.
(2) We are moved to feel for fictional characters.
It's not straightforward to attribute this claim to Hume, either, because many times when Hume is talking about our responses to fiction, he is primarily concerned with our responses to the eloquence of the discourse, which is distinct from our responses to the characters. But he does occasionally say things that are naturally interpreted as at least including responses to characters. For instance;
A spectator of a tragedy passes through a long train of grief, terror, indignation, and other affections, which the poet represents in the persons he introduces. As many tragedies end happily, and no excellent one can be composed without some reverses of fortune, the spectator must sympathize with all these changes, and receive the fictitious joy as well as every other passion. (T 184.108.40.206)
Thus when we watch a tragedy, we certainly experience responses to it; these Hume attributes to sympathy with the passions as represented in the characters of the tragedy. This sympathy is much the same kind as that which we experience with real people.
A complication is that Hume is insistent that the power of a fiction to elicit response is closely tied to belief in existence -- it is the latter (truth or reality, as Hume occasionally calls it) -- that gives force enough to the idea that it calls up significant passional response. But he does not quite commit to this being universally true, and in at least one example identifies something that is put in the place of this belief in order to fulfill the same function:
Poets have formed what they call a poetical system of things, which though it be believed neither by themselves nor readers, is commonly esteemed a sufficient foundation for any fiction. We have been so much accustomed to the names of MARS, JUPITER, VENUS, that in the same manner as education infixes any opinion, the constant repetition of these ideas makes them enter into the mind with facility, and prevail upon the fancy, without influencing the judgment. (T 220.127.116.11)
Similarly, tragedians will sprinkle their works with bits and pieces of truth, without doing so in a way that suggests that the the rest is true, in order to siphon off, so to speak, some of the forcefulness of the belief to evoke an emotional response.
(3) We know that fictional characters do not exist.
The same passages serve to establish the third point, namely, that this can all happen in contexts in which we recognize that we are considering something that does not exist. Neither the poets nor their readers think that Mars and Jupiter are real; the tragedians do not even pretend to be giving us the truth.
Thus we can find in Hume some reason for affirming all three prongs of the paradox. Are there reasons to think that this is merely apparent, however? There are a few passages which you could point to if you wanted to argue that Hume in fact rejects (2). For instance, Hume thinks there is a palpable difference in the 'feel' of ideas that are connected with the belief in existence and those that are not. One passage in particular is especially relevant to our topic:
If one person sits down to read a book as a romance, and another as a true history, they plainly receive the same ideas, and in the same order; nor does the incredulity of the one, and the belief of the other hinder them from putting the very same sense upon their author. His words produce the same ideas in both; though his testimony has not the same influence on them. The latter has a more lively conception of all the incidents. He enters deeper into the concerns of the persons: represents to himself their actions, and characters, and friendships, and enmities: He even goes so far as to form a notion of their features, and air, and person. While the former, who gives no credit to the testimony of the author, has a more faint and languid conception of all these particulars; and except on account of the style and ingenuity of the composition, can receive little entertainment from it. (T 18.104.22.168)
Thus Hume thinks that it matters materially whether you are reading a work as history or as fiction: it doesn't change the ideas, but belief involves a greater forcefulness of ideas, and thus we 'enter deeper' into the passions and concerns about which we are reading; without this it is primarily "the style and ingenuity of the composition" that elicits emotional response.
However, even this is not quite so straightforward, because Hume does still recognize the phenomenon of "poetical enthusiasm" -- i.e., the experience of being carried away by a fiction, and this does involve emotional responses of exactly the sort we are considering. To be sure, the lack of belief means that the response cannot be as intense, but the poetical enthusiasm can still stand in, as noted above, for the belief, even if it does so inadequately:
The mind can easily distinguish betwixt the one and the other; and whatever emotion the poetical enthusiasm may give to the spirits, it is still the mere phantom of belief or persuasion. The case is the same with the idea, as with the passion it occasions. There is no passion of the human mind but what may arise from poetry; though at the same time the feelings of the passions are very different when excited by poetical fictions, from what they are when they are from belief and reality. A passion, which is disagreeable in real life, may afford the highest entertainment in a tragedy, or epic poem. In the latter case, it lies not with that weight upon us: It feels less firm and solid: And has no other than the agreeable effect of exciting the spirits, and rouzing the attention. (T 22.214.171.124)
Whether or not we recognize that a character exists, in other words, will affect the kind of emotional response we have; but even if we take the character not to exist, we can still have some kind of emotional response: an excitement of interest, an arousing of attention. Poetry and madness are alike in this kind of emotional response; poetry is just much more restrained.
Another claim Hume makes is relevant to (3):
It is however certain, that in the warmth of a poetical enthusiasm, a poet has a counterfeit belief, and even a kind of vision of his objects: And if there be any shadow of argument to support this belief, nothing contributes more to his full conviction than a blaze of poetical figures and images, which have their effect upon the poet himself, as well as upon his readers. (T 126.96.36.199)
Poetical enthusiasm, therefore, involves a "counterfeit belief" or even (at its strongest, and no doubt this is why Hume calls it "enthusiasm", which in his day doesn't mean just eagerness but suggests the ecstasy of the visionary, someone who feels that God is speaking to him or sees visions) "a kind of vision" of it. As he noted in passing above, the emotion given by poetical enthusiasm is associated with a "phantom of belief or persuasion". Thus one might also argue that Hume only accepts a qualified version of (3).
Is there any way to nuance Hume's acceptance of (1)? It is true that he never says, flat-out, that passions connected to the false judgment that something exists are always unreasonable, so perhaps this gives some wiggle-room in interpretation. But it is very natural to interpret this as being the import of Hume's claim, particularly when one compares it to what he says about existence and nonexistence elsewhere (for instance, in his discussion of analogical reasoning). And he never gives us any further criterion on the matter.
So after this crude, preliminary survey of Hume's account of poetry, we seem to have the following possibilities for how Hume's theory of poetry could be interpreted with respect to the paradox of fiction.
[A] Hume is in fact committed to all three claims, and thus to the conclusion that our emotional responses to fictional characters are irrational. As Hume is not afraid in other contexts to attribute irrationalities to the human mind, one cannot rule out the possibility that he would be perfectly happy with this.
[B] Hume qualifies (2), so that while we have emotional responses to fictional characters in some sense, this is not necessarily in the sense that is intended in (1). Hume repeatedly indicates that the feel of the passions in the poetic case is very different from that which is found elsewhere, so the kind of passional response we have in this case is not a response-to-truth passion but a response-to-fiction passion. This alternative is related to quasi-emotion or pretense solutions to the paradox of fiction. The most obvious problem with such accounts is that the 'quasi-emotions' or 'pretend emotions' seem to work exactly like emotions. But Hume doesn't run into this problem: we can have passional responses of some kind to fiction, but the kind of passional response we have is just a palpably different kind of passional response than the passional response we have to reports of real events, and the difference is precisely that the latter is connected with judgment of existence and the former is not -- and Hume is only committed to (1) if we are, in fact, talking about judgments of existence.
[C] Hume qualifies (3), so that we can in fact have something like belief that a fictional character exists, and it is this that brings in the emotional response. This gives us a quasi-belief or illusion solution to the paradox. In the grip of 'poetical enthusiasm', we have a 'counterfeit belief' -- something that is not a belief in the strict sense, but sufficiently belief-like to have a similar effect. Thus, for instance, Hume thinks that belief is just a very vivid, vivacious, forceful idea; but vivacity is a spectrum, and so there can be cases where the vivacity is not enough for belief, but still enough to be significant, particularly given that vivacity is Hume's means of explaining how ideas give rise to emotional response in the first place. We can have an "illusion" or "kind of vision" that has similar effects to believing that something exists. Of course, Hume is very clear that this is only partial; there is a definite difference between the two cases. But he does often talk as if there is a kind of illusion arising from poetry that can give rise to passional response.
All of this is, as noted above, somewhat rough and preliminary, so there may be evidence in Hume relevant to deciding which of these three interpretations is most appropriate to Hume. By looking at how the paradox of fiction can arise in a real-life case like that of Hume's theory of belief, one gets a better sense of the topics and ideas with which the claims involved may be associated.