Does this cause any concrete problems in any user agent?

Not that I'm aware of today. (Caveat: I've only used Phabricator in desktop webkit-based browsers.)

I'd be cool with moving to <code> semantically anyways, but imagine this is a bit harder since we'd have to invalidate all Remarkup cache?

Invalidating the cache is trivial, but I think <code> is used to style code blocks and has a bunch of different rules (e.g., block formatting) already associated with it.

If you want to do the legwork for this you can nuke the caches by bumping PhabricatorMarkupEngine->$version.

This doesn't seem worth changing to me until there's some actual problem, but I don't think changing it will do anything bad.