Bangladesh made 306 for seven declared after winning the toss. The Invitation XI were 21 for one at the close.
Tamim, a veteran of 51 Tests and the team's all-time leading run-scorer, made five runs before leaving the field in the fifth over.
Team manager Minhazul Nanna said the batsman had suffered a strained right thigh muscle. He left the field as a precaution and his condition would be assessed on Friday following a scan.
Bangladesh are already without leading all-rounder Shakib Al Hasan, who is being rested from the two-Test series against South Africa which starts next Thursday.
Mominul Haque, captain Mushfiqur Rahim and Sabbir Rahman made half-centuries for the tourists but Mahmudullah, who is hoping to make a comeback to the Test side after being dropped earlier this year, was out first ball to 19-year-old fast bowler Michael Cohen, who took four for 23.
Scores in brief: Bangladeshis 306-7 declared (Soumya Sarkar 43, Imrul Kayes 34, Mominul Haque 68, Mushfiqur Rahim 63, Sabbir Rahman 58 not out; Michael Cohen 4-23). SA Invitation XI 21-1.
First Published: September 21, 2017, 11:32 PM IST