"What" is not needed in either.
She does nothing but study English.
She is doing nothing but studying English.
Student or Learner
- What she does is nothing but (to) study English.
- What she is doing is nothing but (to) study / studying English.
I would like to ask if I should put 'to' or not. Are both OK?
I disagree with Mike about the requirement for "What". The use of the word "is" after "does" and "doing" means "What" is required. Otherwise, you are left with "She does is nothing but study English" and "She is doing is nothing but study English", neither of which is grammatical.
Mike's revised sentences are, of course, absolutely fine but perhaps should have said that both "What" and "is" have been omitted in his versions.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.