TAIPEI—Voters handed Taiwan’s ruling party a rebuke in local elections, raising doubts about President Tsai Ing-wen’s prospects for re-election and potentially heralding a shift in the island’s tense relations with China.

Ms. Tsai’s Democratic Progressive Party suffered its worst electoral setback in at least a decade in Saturday’s elections, losing a slew of large and small mayoral and county executive races, even in its traditional power base in the island’s south. The opposition Nationalist Party picked up many of the critical...