‘Any’ can be used as an adverb to mean ‘at all’ or ‘in some degree’. In the English Vocabulary Profile, ‘any’ is listed as
ADVERB B1 used in questions and negatives to emphasize a comparative adjective or adverb
*’any more’ is listed at A2 to talk about an activity that has stopped.
Therefore, you can ask “Do you feel better?” or if you want to emphasize ‘better’ meaning ‘even a little bit’ then you can use ‘any + comparative ‘better’: “Do you feel any better?”