If it doesn't simply peel up to your liking, leave the piece out in the bright sun for a few minutes, with the back facing sun. Be very attentive, even on a sub-freezing day the sun can rapidly heat dark surfaces to well over 100F. Don't even try to preserve the existing back cover, it's impossible.
Be very careful with solvents, some of the plastic wraps on cheap moulding might dissolve, and polystyrene moulding will definitely dissolve within a few seconds. And solvent can leach colored pigments in the frame, back cover, and matte into the artwork.
Usually you can pick up tape residue by poking at it with masking tape, warmth helps.
If complications develop, seek help on thegrumble.com. They just love to talk about this stuff!