How can I paint faux paneling?

You could definitely paint the paneling, and that would be easier than removing it.

Yes you can.

Make sure the paneling is clean and dry. Prime with a good stain blocking primer like Kilz. Then paint like painting regular walls. Acrilyc latex paint will work. If the paneling has grooves you can fill them with non shrink drywall patching compound. Just put on enough to fill the grooves. Let it dry overnight. Next day use a damp sponge to smooth the patch. Then prime and paint.

Painting is no issue but realize it will look like painted paneling if you just paint it, which is not a bad look. But you could also fill in the grooves and create a smooth surface - even with filling them in - there still might be able to see the grooves. So - I would either remove it if you don't want the grooves or just paint - make sure to use a good stain blocker/primer - XIM has a ton a great products - just google them to see which one would be best. Then just prime and paint - if you do White or Off-White, I think this would be a great look.