Some people have very strong opinions on this but mine is that, as is often the case, it depends. Not all dust covers are affixed to the turntable in the same way and left on during play some can transfer vibrations back to the phono cartridge. Others, such as the dustcover on my VPI TNT III, are completely independent of the plinth, so that's less likely to be an issue.
Back in the '80s, I had a Denon DP-80 turntable in a VPI HW-9 two-arm base. One problem I had with it (and that really took some sleuthing to resolve) was that static electricity from the dustcover was enough to decrease the VTF of either arm. So I always used that with the cover off.