Hair dryer isn't going to be helpful. The tape is made to withstand high heat, much higher than a hair dryer. It's called "thermal" tape after all. And it's nothing like the adhesive.
If you can get the strips off the sheet, you might be able to scrape off the tape by laying the strip diode-side down on a soft surface (towel) and scraping with your thin bladed taping or plaster knife. But you could also just leave it on and re-mount them on SolSinks using screws. A bit redundant but wouldn't hurt the strip any.
The real issue is going to be getting the strips off the sheet without flexing them so much you break the circuits in the PCB. My advice would be to save the sheet for other uses - mother/clone space, kitchen greens garden, reef tank light, etc. and build your rack with new strips. -b420