I have seen a few questions or answers with "Hello!" or "Hi!" in them. Is it safe to ignore these or should I edit them out? And, since they are new users, is there a tactful way to respond while editing them out? They shouldn't feel they have done something terribly, horribly awful but I don't know how to do that without leaving a comment explaining myself every time.
It can be, but only if you have another reason to go in and edit -- salutations, greetings, and taglines (thanks!) are always candidates for removal if you have reason to edit a post.
I would not, however, edit a post with the sole reason of removing these things.