In the dictionary, run after means to seek someone's attention for courting. It also means to chase someone.
Is the sentence given below correct, keeping in mind the various meanings of the phrase "run after?"
Your teacher will keep you run after him for notes till the day before the exam.(to mean the[STRIKE] dilly dally of[/STRIKE] teacher is dilly-dallying in providing notes)
thanking you in anticipation
Firstly, in the instances where a verb follows "keep," I can't think of any examples where the second verb is not in the gerund form, as in:
"Keep going." (= continue)
"He kept asking me to come."
"You just keep me hanging on." (song lyric)
So your sentence would become "your teacher will keep you running after him for notes till the day before the exam." An English speaker would recognize what you're saying with this sentence, and it sounds grammatically correct, I'm just not sure how natural it sounds (at least to me). When the meaning is that the teacher needs to be consistently reminded to give class notes, I would be more likely to reword it slightly. Using most of the same words you used, I'd say something like "you'll need to keep after your teacher for the notes till the day before the exam," where 'keep after' means that you continually tell someone or remind someone about something they need to do. There are several ways of expressing the same idea using different words, I just wanted to use as many of the same words as possible that you used.
Hope this helps!
(not an English teacher - just a language lover)