This is sort of related to this question, How does Doctor Strange remember previous loops?, but it does not answer what I am trying to find out.
Just re-watched this scene on Blu-Ray, and it's unclear whether or not he remembers the pain from each death, but it is implied that he will be by Dormammu in the following conversation:
DORMAMMU - You cannot do this forever.
DR. STRANGE - Actually, I can. This is how things are now; you and me, trapped in this moment, endlessly.
DORMAMMU - Then you will spend an eternity dying.
DR. STRANGE - Yes, but everyone on Earth...will live.
DORMAMMU - But you will suffer.
DR. STRANGE - Well...pain's an old friend.