SÃO PAULO—A Brazil Supreme Court judge ruled that Congress must open impeachment proceedings against Vice President Michel Temer on allegations he helped conceal the country’s budget deficit, raising the possibility that the country’s top two elected officials could be ousted.

Tuesday’s preliminary ruling must still be affirmed by the full Supreme Court, but it already clouds the question of who would succeed President Dilma Rousseff, who is fighting to stave off her own impeachment. She faces allegations, which she denies,...