I like the err on the side of precaution.... call the vet for advice and see what they say. If they are a good vet they will tell you what to do and not have you come in if its something that you can do.
I know from personal experience that when you have a nail that has been torn down past the "quick," it is VERY painful. If you don't have the courage to clip it off yourself, then take the tortoise to the vet to have it done. Then it would be a good idea for a round of antibiotics because of where the injury is. Lots of germs in the substrate and it might get infected.