OTTAWA—For Justin Trudeau, being elected prime minister of Canada was to have been a homecoming of sorts: a return to the official residence at 24 Sussex Dr., where Mr. Trudeau spent much of his childhood as the son of one of the country’s longest-serving prime ministers, Pierre Trudeau.

But the Trudeaus can’t go home again, they say. The house, apparently, has become too much of a wreck for them to move in. Their decision is turning Canada’s most famous fixer-upper—a 34-room mansion on 2.5 acres overlooking the Ottawa River—into...