The Welsh loose forward sustained the injury late in Saturday's 16-15 second test defeat in Melbourne, as the Wallabies levelled up the series at 1-1 after the Lions had won the opener in Brisbane.
He had a scan on Sunday to determine the extent of the injury and it was ruled that he would not recover in time for the decider.
Irish centre Brian O'Driscoll is expected to take over the captaincy duties while Ireland's Sean O'Brien or fellow Welshman Justin Tipuric will likely take Warburton's place in the starting XV.
"The British & Irish Lions confirmed today that captain Sam Warburton would play no further part in the tour after suffering an injury during the second Test," the Lions said on their official Twitter feed on Monday.
"He will remain with the tour party for the third test in Sydney on Saturday."
Warburton, who was assisted from the field after 68 minutes, gave his best performance of the tour to date as he helped the Lions gain a vital edge at the breakdown against the Wallabies.
The hosts scored the only try of the match after Warburton left the field to overturn a 15-9 deficit with a converted Adam Ashley-Cooper effort in the 75th minute.
"It is incredibly disappointing to be ruled out of the tour through injury as we head to Sydney for the deciding match," Warburton said.
"I wish the team all the best and hope I can play some part in the build up this week. I am confident that the boys can finish the job off and secure the series win."
The Lions are looking for their first series win since 1997.