Not under the Constitution. But when a vote of no-confidence passes the House of Representatives - or a confidence vote is rejected - the prime minister must either dissolve the lower house within 10 days or resign along with his or her cabinet ministers.
The Japanese Constitution says that the prime minister be a member of the National Diet and be designated by Diet resolution after receiving a majority of votes cast in a Diet election. A new prime minister must be appointed by the Emperor, although this is a formality.
In case the cabinet resigns or when the Diet is convened following an election, the designation of a prime minister takes precedence over all other Diet matters.
Historically, the person named to become prime minister has generally been the leader of the biggest party in the lower house.