What does the inspector accuse David of doing in the Chrysalids?

Answer 1

The inspector confronts him about Sophie, telling him that it is against the rules to hide deviants, etc... so perhaps that: hiding Sophie.

He never gets accused of being a telepath, because he has already run away at that point.

Answer 2

The inspector accuses David of being a blasphemer and having the ability to communicate telepathically with others, which is seen as a mutation and therefore a threat to the purity of Waknuk society in "The Chrysalids."