Ok guys

I believe it depends on how solid your turntable is in correlation to how much it will affect the sound.  When you have the cover up it has to gather more sound waves like a sail on a boat. The best would be the kind that does not touch the turntable. If you notice really nice turntables do not have covers attached to the table itself. Cover off while playing is the best. If you want to test it. Have some one listen to the speakers while you talk into the cover with the player on.  I keep my record player in a cabinet sitting on a separate plinth with dampers so the sound from the speaker does not affect the music. Like someone said about a separate room. I just isolated it from the affects of the room. 

crustycoot

I’ve seen low mass arms so attracted by static from the dustcover that they are lifted right off the record! Stands to reason that less static can also attract the arm, even if not enough for full lift-off.

B-i-n-g-o! Give this man a gold star! I have deliberately avoided commenting on this thread, and instead waited for someone else to make this observation. The phenomenon was a particular problem for me back in the early ’80s when I used a Denon DP-80, which I had installed on a nifty two-arm base made by a new company based in the oddly named "Ozone Park" called VPI.

So the best answer to the OP’s question is - as is often the case in audio - "it depends."

Not playing records, dust  cover on.

Playing records, dust cover off.  

I can hear feedback at significant volumes with cover on, so easy choice.

Wolf how is a dust cover going to be a bass trap? Do you know what the wavelength of 20 Hz is?? The wavelength of your dust cover is 800 Hz or so. But none of this determines how your dust cover resonates. If you have a flimsy dust cover directly attached to the same platform your tonearm and bearing are mounted to and that platform is not sufficiently heavy, stiff and non resonant. You might indeed have a problem. Correctly made dust covers and turntables like the SOTA and dust covers that are not directly attached to the turntable improve the sound by attenuating the noise (music) in the room by about 20 dB depending on the cover, like having hearing protectors on. 
Now If you have static in your dust cover lifting your tonearm you need to throw the whole mess away and start over. Better yet switch to digital only. But then I guess you'll get paranoid about something else.
If you really think your dust cover causes sonic deterioration you also need to throw the whole mess away and get a decent turntable. Protecting your records is the first priority, sonics second. But Like I have said repeatedly all the turntables I have had in the last 40 years have sounded better with their dust covers down. I guess I know how to pick them or is it just dumb luck. 
I love you guys who complain about static and spend thousands of dollars on record cleaning junk. I use conductive sweep arms and dust covers, get better sound and I never clean a record I have owned from new. Never have to. The secret to cleaning records is don't let them get dirty. 

dust covers that are not directly attached to the turntable improve the sound by attenuating the noise (music) in the room by about 20 dB depending on the cover, like having hearing protectors on.

Wow 20dB 🤯.....here's the miracle sound-proofing material the world has been waiting for.......1/8 inch acrylic 🥳
And no proof required...🤣
Wait till I notify that village looking for it's inhabitant.....that he's been found 🤥