When I leave my current apartment, I will have quite some holes in the walls to fill due to numerous shelves and such. I know how to remove wall plugs, but I was wondering whether you should do it in the first place. I'm guessing it depends on the type of wall and the type of plug.
As the owner of a drywall repair company, I have fixed thousands of holes like this, and I prefer to slightly tap them in, or drive them in until just slightly recessed. Then coat them like usual. The reason is if you fill an open hole the mud tends to droop back out as there's no backing. And if it's all slightly recessed it fills better and tapers out better. But you can fix it just by filling it, then tapping it in slightly, when it starts setting up, or just sand it down, then coat it some more.
If you want to see an actual video of this being done, contact me in private. I post this because it's often easier to figure out how to do these things when you can see it happen, as verbal only goes so far.