How can I repaint an antique piece of furniture that was varnished?
You can paint over varnish, but you will need to thoroughly sand and degloss the existing finish first.

I use medium grit sandpaper followed by fine, then wipe with solvent to remove all dust, dirt etc., then paint with high quality oil-based paint using the very best paint brush I can afford. Just finished some old varnished cabinets and they look great now, mirror smooth like glass.