"Sreesanth has a left side strain and he will require two weeks to recover from it," cricket board secretary Niranjan Shah said in a statement.
The bowler will be replaced by the uncapped Manpreet Gony, who impressed in Indian Premier League recently.
Gony's inclusion takes the number of uncapped players in the 15-member Indian squad to three with Pragyan Ojha and Yusuf Pathan being the other new faces.
Pakistan is the third team involved in the Bangladesh tri-series to be played from June 8 to 14 in Dhaka.
Sreesanth, however, will be available for the Asia Cup to be played in Pakistan from June 24 to July 6.
India are already without star batsman Sachin Tendulkar who will sit out of both the tri-series and the Asia Cup due to a groin injury.
India squad for the Bangladesh tri-series: Mahendra Singh Dhoni (capt), Virender Sehwag, Gautam Gambhir, Robin Uthappa, Yuvraj Singh, Rohit Sharma, Suresh Raina, Irfan Pathan, Yusuf Pathan, Piyush Chawla, Pragyan Ojha, Manpreet Gony, Rudra Pratap Singh, Ishant Sharma, Praveen Kumar.
Coach: Gary Kirsten