Zimbabwe, returning to Test cricket after a six-year exile for political reasons, beat Bangladesh by 130 runs in a one-off match.
The home side also won the One-Day International series 3-2, Bangladesh's first limited-overs series defeat by Zimbabwe in six years.
"Shakib and Tamim and have been removed from their posts as they failed to lead the team to success in Zimbabwe," BCB spokesman Jalal Yunus told reporters after a meeting of the Board of Directors.
Shakib was appointed captain on a permanent basis in February and was due to lead the team until Bangladesh's home series against Pakistan in December.
Earlier he led the side on an interim basis in the absence of injured Mashrafe Mortaza and almost single-handedly guided Bangladesh to an unprecedented 4-0 win over New Zealand in a one-day series last year.
He came into conflict with officials when he questioned team selection before the Zimbabwe tour and received a written caution.
Bangladesh's next opponents are West Indies who they host in a Twenty20 International, three one-dayers and two Tests starting in October.
