To answer your question directly, "questions" is plural and so should always have a plural verb ("are"). However, there are two other issues with your sample answer:
- Your are eliding the main subject and verb from your answer. This makes the sentence potentially hard to read, and ungrammatical in a formal setting. (In some kinds of writing/speaking it would be acceptable).
- You need a comma after the quotation.
I learned that silly questions, like "What did you learn from doing this task?", are a complete waste of time.
The rule for the subordinate parenthetical clause in the middle of your sentence is that it can be removed without affecting the meaning or structure of the sentence.
I learned that silly questions ... are a complete waste of time.
This makes it more obvious that you need to use the plural verb "are". It also makes it clear that your subordinate clause should be framed by two commas, and not just one.