Theekshana strained his right hamstring while sliding near the boundary and received treatment from the team physiotherapist. He bowled three more overs before being helped off the field during Sri Lanka's thrilling two-wicket win on Thursday.
Sahan Arachchige has been called up as his replacement for Sunday's final in Colombo, where Sri Lanka will aim to retain their title.
Theekshana will "definitely be ready" for next month's 50-overs World Cup in India, which starts on October 5th, according to Professor Arjuna de Silva, chair of SLC's medical committee.
"The MRI scan is showing a tear - not a big tear," he was quoted as saying by ESPN Cricinfo.
"Clinically, Theekshana is okay. He's moving around and not feeling too much pain.
"If we didn't have a World Cup coming up, we would have somehow tried to get him ready for tomorrow's match. But we don't want to take that risk."
Washington Sundar is expected to join the India squad as cover for Axar Patel after the all-rounder sustained an injury in Friday's six-run loss to Bangladesh, cricket website Cricbuzz reported.
Sundar was named in a second-string India squad for the Asian Games in the eastern Chinese city of Hangzhou, where the men's competition starts on September 27th.
He is expected to rejoin the Asian Games camp after the Asia Cup final, the Cricbuzz report said.
