The part of the Supreme Rule which expresses the claim of this Virtue, is this: We must speak the truth: which may be farther unfolded, by reference to the origin of the principle, in this manner: We must conform our language to the universal understanding among men which the use of language implies.
As he puts it later, "a Contract to speak the Truth is implied in the use of Language" (EM §216), and by making an assertion one is implicitly recognizing some kind of broad right of those addressed to know the truth. The subjective disposition described by this Idea, of course, is the Virtue of Veracity, and the duties associated with it follow from this basic rule, understood broadly.
The dispositions associated with this Virtue are as you would expect. Fidelity or Good Faith is conformity of action to engagement, and if love is added to it, it becomes Loyalty. Freedom from fraudulent disposition is Probity. The person with Integrity has Simplicity or Singleness of Heart.
Obviously lying is a violation of Veracity, but so is promise-breaking and contract-breaking. We not only have the duty to avoid deceitful behavior but to cultivate a hatred of duplicity and a desire to be honest with others, and thus, as with the other Ideas, have the concomitant duties of moral culture. Thus our actions should not just be honest but convey the importance of honesty, so we have a duty not only to be truthful but to convey the "Spirit of Truth" (EM §220) in our truthful actions. Deceitfulness must be made unnatural to us.
There are Cases of Conscience for all kinds of duties, but the appearance of conflicting duties combined with the apparent clarity of the conceptions associated with the Idea mean that duties of Veracity are particularly likely to be considered in this context. Common understanding is a major issue here. For instance, a promiser should interpret his promise in the way he thinks the promisee is likely to take it, because that is the only possible common understanding between them (EM §280). In part because of this, promises are only relative duties created by the people involved; the promiser thus is not bound to a promise if released by the promisee. Immoral promises should be broken, but this does not do away, Whewell thinks, with the relative duty between promiser and promisee -- promising something immoral creates a conflict of duties for oneself, and promises morally made should always be kept. To take another example, fictions and polite expressions are recognized as such under common understanding, so they are not lies.
Because of the way he understands common understanding, Whewell has a peculiar view when it comes to what he calls Lies of Necessity, such as when you are faced with a choice of either lying or letting people die; and he regards them as excusable. But the standard of necessity has to be quite strict -- it has to be fear of immediate and inevitable death, although perhaps some non-immediate but inevitable cases might be allowable, and even such cases do not give a blanket permission. Excusable is not the same as blameless or admirable, and where it is admirable, as in some heroic cases, it is so for expressing some other moral principle. Cases may take us beyond our best rules and our usual duties; but they do not take us beyond the standard imposed by the Idea of Veracity.
States as moral agents of course have duties of Truth, as well, for instance, in upholding treaties, but Whewell does not spend an extensive amount of time discussing them. Likewise, while it is true of Veracity as with other cases that the Christian faith intensifies and incentivizes natural duties, Whewell takes the bulk of Christian duties of Veracity to be already also protected indirectly by other duties -- to love one's neighbor, to restrain one's desires, and the like.