Does it hurt a horse to grab its mane?

Not really - if you have long hair grab a HUGE chunk of your hair and pull and you'll note it might be a bit uncomfortable, but not painful. Pulling small amounts of mane will hurt them a little, but as long as you get a good chunk it's fine.

It is not however true that horses "have no sensory nerves in their hair follicles" - that's a myth (and one which makes zero sense as horses have whiskers).

Plenty of mane here!