Your answers to questions, quizzes, and other conversation types won't be lost if you cancel your subscription. The progress with your partner is not attached to any membership.
However, free users can only access the question or quiz of the day. If you miss out on a Daily Question or Quiz, you won't be able to answer it unless your partner answered it on the day it was published.
If you have any more questions on this topic, please let us know with a description of the problem and a screenshot of any error message you see. We'll make sure to investigate further!