Some of the team's players went to a nightclub after playing their first match of the tour and returned to their hotel in the early hours of Sunday, according to local police.
Baty was transferred to the private Nawaloka hospital where he too died on Tuesday afternoon despite intensive care, officials said.
Clem Pirates were in Colombo on tour, having arrived last Wednesday, before facing Ceylonese Rugby and Football Club (CR & FC).
Thomas Howard, 25, and Tom Baty, 26, developed breathing difficulties on Sunday morning after returning from the club in Colombo and were rushed to the main state-run hospital in the capital.
The pair's breathing issues were not rugby-related, the club said, but no further details of the fatal incident were provided.
A second player from a British rugby club visiting Sri Lanka died on Tuesday, two days after being admitted to hospital with breathing difficulties, police said.
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The death was confirmed to News 1st, by Police Media Spokesperson SP Ruwan Gunasekara.
A British High Commission spokesman in Colombo said both families were being supported, and they were in contact with the Sri Lankan medical services.
Durham County Councillor David Boyes said Durham City RFC was a very well organised, well equipped organisation and oversaw a number of teams.
"They do a lot of good work raising money for charity and introducing people to rugby", he said.
One local report said the family of the dead man, aged 26, would travel to Sri Lanka from Northern Ireland.
She tweeted: "Unbelievably tragic news".