Thinking about peeling up the LED strips, using a solvent would probably make it easier and less likely to damage the circuit board.
A good solvent for most adhesives I have run across is simple straight chain hydrocarbons. Zippo lighter fluid works well. It is a light to medium naphtha. Colman white gas is a light naphtha. That Goo Gone is a medium naphtha and light kerosene plus the fragrance (which you pay dearly for). Lamp kerosene would work I'm sure but will take a bit longer to dry. Longer still would be mineral spirits.
The Ronsonol brand zippo lighter fluid is my personal favorite. It come in a conveniently sized bottle with a small spout that doesn't let out too much at one time. It seals well. It is cheap, usually under $3USD.