Yup, it hurts! Well, the bulk of the pushing doesn't hurt itself but as said above it's not called the ring of fire for nothing! haha.
That being said...I had a local and episiotomy (tiny one) with my first due to me giving up and not pushing anymore and the vacume extractor thing going in was the thing that hurt the most!! I didn't notice her crowning as it was so quick and I guess the local was helping (was given the local so I didn't feel the cut). then with DS1 OMG that stung, then AMAZING relief once his head was out and it was almost over. DD2 (twin A) I breathed her out, involuntary pushing and I honestly don't remember it hurting at all. DS2 (twin B) yeah, I felt that, but that was ALOT more forced, as I started pushing him out before he was really ready and I wanted it to be OVER already.
Sorry for the novel!! Basically, yup it can hurt ALOT as bubs is crowning, but it's short lived (generally) and the relief after is wonderful!