Hair dryer and flat scraper. The glue will give way with some heat and just ease the scraper under and gently pull away the strip. Keep the warm air pointed towards the gap to keep the glue softened.
You'll get some glue residue and how you remove that depends on whether you want to replace the moulding or not. The disk-type (nylon) brillo pad drill attachments seem to work quite well without damaging the tube surface.
Generally though, once the glue is on there, it's difficult to get a pristine finish again, so consider what you're going to do before you remove the old one..