If you keep his nails as closely trimmed as possible so there's really not much to go after with his teeth, that may discourage him. Some parents have made up a "recipe" of a sour and or peppery liquid and dipped their child's fingers in it; many have reported that this simple behavioral conditioning technique works well. They have made it liquid enough so that it is like dipping fingers in water, yet potent enough so that the tongue responds immediately to the taste sensation. Obviously one needs to test it out so as not to shock the child's system.
I think calmly taking his fingers out of his mouth and then taking that hand and leading him somewhere else is OK. I would be cautious not to follow the finger removal with a fun activity since he might become conditioned to think that biting his fingers will lead to a special time with you.