The CM11 armoured vehicle carrying five soldiers was returning to base when it fell into the Wangsha river in Pingtung county in the south of the country.
The driver and tank captain survived.
The drill was part of preparations for next week's annual Han Kuang military exercise. President Tsai Ing-wen has sent her condolences.
The CM11 was returning from a firing test at about 10:30 local time (02:30 GMT) when an apparent mechanical failure in one of the tracks left the driver unable to make a required left turn, the army said.
The driver was able to escape with light injuries.
The four others were pulled unresponsive from the vehicle. The captain was revived and later taken to a military hospital in neighbouring Kaohsiung city.
The soldiers reportedly belonged to the Eighth Army Command.
President Tsai demanded a speedy investigation into the cause of the incident.
She is scheduled to preside over the military exercises, which simulate possible attacks by Beijing.
A Taiwanese armoured vehicle has plunged from a three-metre bridge in heavy rain after a military drill, killing three soldiers, officials say.