MEXICO CITY—A taped telephone conversation between a person believed to be a federal congressman and a drug lord has raised new questions about the extent to which Mexican drug gangs have bought off politicians and whether the politicians, if exposed, can even be punished.

Julio César Godoy, a newly elected lawmaker from the leftist Party of the Democratic Revolution, failed to turn up at Mexico's Congress on Friday, a day after a radio station broadcast what it said was a conversation between the congressman...