He has a____tongue; his pinching sarcasm has______ everyone who has come into contact with him.

Go for the second blank first. What is sarcasm? It is saying the opposite of what you mean. One gets sarcastic in order to mock or insult others. Take an example of sarcasm: You meet an ugly girl. She asks you to accompany her. Repelled by her ugliness, you tell her sarcastically, I dont go around with beautiful (obviously, you mean ugly!) girls.
Such a statement pinches the girl. A girl whom you have treated with such pinching sarcasm will never be friendly or sympathetic to you in future. You have thus alienated her. Those who are thus pinchingly sarcastic have a vitriolic tongue. That is, their words are full of bitterness and hate, and so cause a lot of distress and pain.