Will a hermit crab survive if the pincher comes off?

It would depend on why it came off, and that question can be hard to answer. If it came off because of stress, the crab is usually too far gone to recover, and will keep losing limbs until it dies.

If it came off because of a fight, the crab has a good chance of living as long as the environment in your habitat is suitable for the crab to recover in. This means deep, moist sand, proper heat and humidity, etc.

If it came off because the crab was stuck somewhere and broke it in attempt to free itself, the crab will probably be just fine. See above paragraph to make sure you have decent conditions.

If the crab lost it in a bad molt, it will probably survive. See two paragraphs up.

As long as the crab has one claw, it will eat just fine, and will regrow it in the next molt so long as your conditions allow the molt to be successful.