VIENNA—Austrian Chancellor Sebastian Kurz, one of Europe’s most popular conservatives, was ousted by a no-confidence vote following a secret-tape scandal that drove his nationalist coalition partner from power.

The loss on Monday for Mr. Kurz, at 32 years old one of the world’s youngest government leaders, came despite his Austrian People’s Party’s victory in European elections—a result that suggested he could easily win the snap election now expected in early September.

The...