The photo - known as "the Roaring Lion" - was taken by Yousuf Karsh shortly after Churchill gave a wartime speech to Canada's parliament.
A staff member at the Château Laurier hotel first noticed the photograph had been replaced on 19 August.
Police are investigating the incident.
The photograph is one of the most iconic ever taken of Churchill, and shows the leader on Parliament Hill moments after Karsh famously took a cigar out of Churchill's mouth.
"I held out an ashtray, but he would not dispose of it...I waited; he continued to chomp vigorously at his cigar. I waited," Karsh later recalled. "Then I stepped toward him and, without premeditation, but ever respectfully, I said 'forgive me sir' and plucked the cigar from his mouth."
By the time Karsh returned to his camera, he wrote, Churchill looked "so belligerent he could have devoured me".
"It was in that instant that I took the photograph," he added.
The Château Laurier first realised something was amiss on Friday, when staff members noticed that the photograph's frame didn't match that of other images by Karsh in the same room.
After investigating and conferring with Karsh's estate, the hotel determined that the photograph had been "replaced with a copy of the original", the hotel said in a statement to local media. It is unclear how long it has been since the original image was taken.
"We are deeply saddened by this brazen act," the hotel's general manager, Geneviève Dumas, said in the statement. "The hotel is incredibly proud to house this stunning Karsh collection, which was securely installed in 1998."
The hotel is operated by Toronto-headquartered Fairmont Hotels & Resorts, a subsidiary of AccorHotels.
The case has been reported to Ottawa Police, which has launched an investigation into the missing photograph.
Karsh, who died in 2002, is considered one of the famous portrait photographers of the 20th Century, with other subjects ranging from Queen Elizabeth II, Nelson Mandela and Martin Luther King to Fidel Castro, Muhammad Ali and Andy Warhol.
He was a long-time resident of the Château Laurier, from where he operated a studio between 1972 and 1992.