"He got two fractures in his hand. He would not be able to continue here and also for the next three to four weeks," team manager Khaled Mahmud told from Dubai.
Tamim was hailed as national hero after he returned to bat with his injured hand for the final wicket during Bangladesh's 137-run win in Saturday's opening match against Sri Lanka.
The left-hander received a blow to his wrist in the second over of Bangladesh's innings when he was hit by a short ball from Suranga Lakmal.
He left the field for hospital for a scan, and when he returned to the ground with a sling on his injured hand his Asia Cup was thought to be over.
But the opener stunned all by striding out to bat after Bangladesh lost their ninth wicket for 229 runs.
He fashioned a hole in his glove in order to fit his taped wrist, and used one hand in facing a single delivery to give the strike back to Mushfiqur Rahim, who was batting on 112.
Mushfiqur then scored 32 off 16 balls alone in their 10th-wicket partnership, guiding Bangladesh to 261 runs before he was dismissed in the final over for 144.
Bangladesh later bowled out Sri Lanka for 124 runs to record their biggest win on foreign soil, making a giant leap towards the Super Four round.
Bangladesh will face Afghanistan in their second match of the tournament in the United Arab Emirates on Thursday.