What is the best way to remove a wall anchor?

I removed an old mirror from a bathroom and am installing a slightly smaller one, so the anchors are in a spot which will be very noticable if I do it incorrectly.

Depends on the type of anchor you have used. All the above answers are correct but there are different ways to remove different types of anchors. If it was plastic with a screw for expansion, you may be able to remove the screw and back the plastic out, if it had ridges and you used the screw almost out of the plastic to loosen it up a bit.
If it was a toggle type, the one with the long screw and the expanding wings on the end that then clamp as you tighten the screw, just back the screw out and he toggle will fall behind the wall. If it was like plastic, only metal and had a sleeve that drew up either inside the drywall or on the other side of the wall, same thing--take out the screw and the rest of the sleeve should fall.
No matter what you will need drywall compound, several sandings, and repainting.

Remove the old anchors patch the holes and start over with new anchors. I hope this answered your question.

hmmmmm! I'm trying to visualize your predicament. So, your mirror is smaller than the area covered by the larger mirror. I can see other problems, too. But, for now, I would push the wall anchor through and let it drop behind the sheetrock.
I would plaster the hole. This will show up of course, but, maybe you can put a stick on hanger over the hole and hang an objet d'art over the area. You'll have to do the other side the same to give it balance.
Thanks

depending on type you can try to pry out with a small scew driver but allot of times we have to pound them just under the surface and patch over them.

I usually just back them out with a screw driver, but there are ones that won't work with. Either way you are going to have to spackle over the holes!