The Yarmouth Navigator, a former Navy minesweeper and patrol boat, was being moved to a new mooring after a campaign to save it which lasted several years.
Rescuers were searching for one missing person after three people were saved from the waters of Plymouth Sound shortly after 6.30pm.
A major search and rescue operation was launched, with officers from Devon and Cornwall Police, crews from Brixham Coastguard, a search and rescue helicopter and RNLI lifeboats involved.
The vessel is understood to have been in the process of relocation from its former mooring in Noss Marina, on the Dart, to Plymouth.
The Yarmouth Navigator was one of around 5,000 ships that participated in the Normandy landings in June 1944 and it is listed by the National Historic Ships Committee on its register of vital ships. Unlike listed buildings, there is no official protection for ships.