There's some interesting meanings in the two sentences you posted. "May I have some tea?" is a polite way to ask for tea, true, but the connotation means "may I have some of the tea you're having?" If you instead said, "May I have any tea?" you seem to imply that you don't expect to get the tea your host drinks, but instead some other tea available -- probably cheaper and lower quality. Asking for "any tea" would be expected if you were an unwelcomed, or unexpected guest.

Mehranyazdi1980, We tend to use "some" instead of "any" when we expect a positive answer to the questions we are asking or when we are offering or requesting something that is already there. May I have some tea?... instead of... May I have any tea? Would you like some tea?