How can I remove ornamental scrollwork from dresser drawers?

I would gently go around the piece with a paint scraper until it lifts up.

If it's solidly glued, you might try a chisel. I'd try to part it a bit higher than level, so you could later sand the last bit down and not end up denting your surface.

You could use a heat gun to melt the glue but be careful they get really hot. You dont want to burn it or set it on fire.

Are you sure it's an applique and not hand carved???

Use a hair dryer its not as hot as a heat gun..It just might loosen the glue so you can pry it off.

Try taking a Qtip and either acetone (nail polish remover) or paint remover,and a paint scraper to try to lift it a little at a time as you put the remover around it with a Qtip.Like Ardel above said,Are you sure it's not handcarved on? If it is,you have an older piece on your hands worth restoring.

Hope this helps!