An example of a false analogy between energy and mass would be to assume that since E=mc2, then energy and mass must be identical. Energy and mass are not identical, energy can travel at the speed of light while mass cannot. Analogies should never be mistaken for establishing an equivalence. Not recognizing the misapplication of analogy can be as potentially disastrous as not recognizing a misapplication of logic.
It's certainly true that analogies should not be confused with equivalences or identities, but it's hard to see how such a confusion is supposed to be an example of a fallacy of false analogy, given that the whole problem is that the analogical character of the relation is not being recognized. It's also not obvious that the proper way to think of the equality in the E=mc
But, as I think I've suggested before, the problem here is not in Wikipedia or its editors but in the fact that standard accounts of false analogy are neither consistent nor well-motivated. Part of it is that standard accounts of almost all the fallacies are patchwork, crazy quilts sewn out of striking pieces of accounts that are not obviously consistent and sometimes are obviously not consistent, because the accounts are built up in the way any folklore is, however philosophical this form of folklore may be.
Part of the problem, however, is historical; we seem to owe our special confusion about false analogy to John Stuart Mill and his readers. Mill, in A System of Logic, doesn't seem to be the originator of the label, but his account of it was for a very long time the most widely read account. his account, however, has a number of idiosyncrasies. On Mill's view there is one legitimate form of analogical inference, which is basically nothing other than an inductive argument to raise the probability that this or that cause is operative in a certain kind of phenomena. In other words, since induction is primarily taken to be about causes, analogy is a form of causal generalization; false analogy is a form of bad causal generalization. We see the residue of this in the fact that a number of late nineteenth century authors treat false analogy as a lapse in causal reasoning, non tali causa pro tali causa. In order to be legitimate, the bare analogical inference is not enough; you need in addition a reason to think that it takes into account all the relevant causal conditions. It is because of this that his description of false analogy almost makes it sound like all analogy is out the window:
There is another, more properly deserving the name of fallacy, namely, when resemblance in one point is inferred from resemblance in another point, though there is not only no evidence to connect the two circumstances by way of causation, but the evidence tends positively to disconnect them. This is properly the Fallacy of False Analogies.
All analogical inference requires that there be "evidence to connect the two circumstances by way of causation." This is an extraordinarily strong condition on analogical arguments -- one can see the reasoning behind it, but very few people think that the field of legitimate analogical reasoning is this sharply restricted. For instance, many people think that there can be legitimate analogical inferences to the relevant causal conditions; these would all arguably be either illegitimate or circular on Mill's account. What seems to have happened, then, is that some of Mill's claims about false analogy continued to be accepted, but detached from Mill's unusual account of analogical legitimacy. The result is incoherence: on almost any newer account the line between legitimate analogy and false analogy is arbitrary because Millian claims about the fallacy are still fairly standard but are unsupported by any of the very distinctively Millian reasons why Mill originally made the claims.
In order to have a good account of false analogy, then, one needs a good account of analogy. And while some accounts are no doubt better than others, there is no general consensus as to the best account of analogy. I've already mentioned Mill's, in which there is one and only one class of analogies that are good. It's interesting to contrast it to another notable historical account of analogy, namely, Hume's. Hume doesn't talk about On Hume's account of analogy, analogical inferences can be stronger or weaker, but there is one and only one class of analogies that could be taken to be candidates for false analogy: those that essentially posit a similarity between existence and nonexistence. In other words, on Hume's account as long as there is any real (i.e., not purely verbal) similarity an analogy has force. (This is actually somewhat important for understanding Hume and his project, but getting into this would be too long a digression here.) I would suspect that most people have a view of legitimate analogical inference that lies somewhere between Mill's minimalism and Hume's maximalism. (Hume has always seemed to me to be obviously right on this point, but I have found that many people do not agree.) The problem with regard to false analogy should be clear enough: even assuming that there is some coherent fallacy here, the slate of possible candidates for examples of such a fallacy varies wildly depending on what, exactly, your view of analogical inference is. Finding even a reasonably uncontroversial example of false analogy will be fairly difficult, given that people don't even agree as to what makes an analogy bad in any sense.
UPDATE: Reworked some of the statements for greater accuracy. Also, it should be pointed out that, despite the label 'minimalism' used here, Mill actually allows quite a bit of room for analogical inference -- what makes him a 'minimalist' is not that he thinks there is very little value to analogical inferences or that they should be used minimally but that on his account all analogical inferences have to meet fairly restrictive conditions to be of significance.