In a conventional DSLR, the mirror needs to come back down between frames so the camera can use its normal autofocus and metering systems, which are located behind the mirror. That matters most in burst shooting when subjects are moving and the camera needs to refocus and re-meter for each shot.

The mirror also restores the optical viewfinder image between frames, letting you keep tracking the subject and surroundings in real time.

If the mirror stayed up for the whole burst, you would lose that normal DSLR viewfinder image, and in most designs you would also lose between-frame AF and metering. Some higher-end DSLRs offer faster burst rates with the mirror locked up or with AF/metering locked from the first frame, but this comes with restrictions.

So yes, keeping the mirror up can improve burst speed, but the tradeoff is reduced tracking, autofocus, and/or metering performance. Mirrorless cameras avoid this specific limitation because they use an electronic viewfinder fed directly from the imaging sensor.