I don't think this is a good idea. It works for other languages because the primary purpose of those languages is current communication in those languages. The primary purpose of Latin and other dead languages is instead to understand written texts.
Given that this is an English site and that the people who primarily come here know English, I can see zero benefit is making the tags Latin. In fact, despite Ben Kovitz' idea of mods tagging posts, one of the main purposes of tags is to allow users to explore efficiently. That purpose is wholly defeated if we resort to esoteric language and possibly made up words for which the ancient Romans had nothing comparable.
Editing this to incorporate Earthliŋ's comments, which I wholeheartedly agree with. I think this would make an excellent compromise:
Something that we did on Japanese.SE is to use English tags, but give a Japanese translation right at the beginning of the tag excerpt (picture). This way the tags are easier to type and recognize, but hovering over them reveals the translation straight away. We could decide whether we want to include pseudo-Latin translations for modern words, but at least they won't feature as strongly and we would have the most useful/widely-known terminology on the tag itself. This might be a "best of both worlds"...?
Edited again just to keep the conversation going. Can someone chime in, one way or another?