I too am experiencing a similar issue with Do Not Disturb using Big Sur 11.2.1. Do Not Disturb will automatically be enabled when I boot up my machine even though I do not have any of the options checked off for Do Not Disturb when opening notification settings in system preferences.
I tried a few other attempts to fix the issue, which I've listed below, but the problem continues to persist.
1. I tried the steps listed in the MacOS subreddit, but no dice.
https://www.reddit.com/r/MacOS/comments/kcvdn4/cannot_turn_off_the_do_not_disturb_on_big_sur_is/
2. I made sure that I didn't have Do Not Disturb switched on on my iPhone and I also turned off the Bedtime iOS feature in case that was the culprit.
3. When trying to turn it off, whichever Do Not Disturb option I select doesn't take effect until I click back on Do Not Disturb as seen in the attached clip.
I didn't encounter the above issue with the release of 11.0. I began to notice it after installing 11.2.1
Here's a short clip of the issue:
https://www.dropbox.com/s/hru99rm6r6w2hq2/Do%20Not%20Disturb%2011.2.1.mov?dl=0