I believe that in many cases methods used for functional equations and inequalities use similar results and some results for functional equations have analogues for functional inequalities. See e.g. the book Marek Kuczma: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality.
So I think that the best solution would be to use the same tag for both functional equations and functional inequalities. Since this might be confusing for some posters (most of the people probably don't check tag-wiki), I propose to keep functional-inequalities tag, but make it a synonym of functional-equations.
Question related to the proposal: Only the users which have some rep in the tag can vote to make that a synonym. At the moment functional-inequalities contains only one question and there is no answer to it. So there are no users that could upvote the proposal to make the two synonymous, so there is a technical problem with my suggestion.
Is there something mods can do about this? Can mods make synonyms without users having to upvote the proposed synonym?
But it was explained in Willie Wong's comment bellow that a moderator can make synonyms, so there is a way to bypass this problem (if we decide to use this solution).