How do you remove black permanent marker from a wooden frame?

Likely wipe with goof off. Beware it also removes paint.

Try WD40

I'd take a q-tip dipped in denatured alcohol and gently wipe towards the center of the mark. Try the alcohol on a small hidden area first to see how it reacts with the finish. If this doesn't work, you may have to sand the area and re-stain to match the finish.. Alcohol does remove permanent marker.

Ignore you have nothing to loose try a magic eraser. It may take some elbow grease.

Bleach takes it out

Nail polish remover.