1. The foreign language _that_ I like best is French. — Correct. "That" can be omitted in this case.
2. David Crystal is the linguist _who_ wrote some fascinating books about the English language. — Incorrect. The word "who" cannot be omitted as it identifies the subject of the relative clause.
3. Can you tell me the name of your friend _who_ I met last night? — Correct. "Who" can be omitted here as it is not essential for the meaning of the sentence.
4. This is the list of verbs _which_ you need to revise. — Correct. "Which" can be omitted as it is not necessary to understand the sentence.