If you stick your finger in a Venus Flytrap's mouth will it bite you?

No, it is just a trap and not a very brutal one, just enough to hold a fly. In order for the plant to hurt you, you would have to leave you finger (or whatever) in the trap until the digestive enzymes have a chance to burn through your epidermis, which would be days.

Also, the plant does not have teeth really. They only can put out enough pressure to crush their prey.