He is suffering from a thigh muscle tear and could be out for a month, jeopardising his participation in three One Day Internationals against South Africa which follow the two-match test series.
Tamim is to stay with the squad in South Africa for rehabilitation.
Bangladesh are already without another of their leading batsmen, Shakib Al Hassan, who asked to skip the tour to rest because of exhaustion.
South Africa won the first test in Potchefstroom by 333 runs on Monday.
(Reporting by Mark Gleeson in Cape Town; Editing by Keith Weir)
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