Now this solution isn't perfect but.... I had the same problem, which is how I found this page! 2 days of scraping a front door, fitted in 2014, facing south, the tape on the door still read, remove within 3 months.... ☺ Now 3 years later, I was faced with the same problem! Tried many solutions but this worked for me. By accident more than design, we had a new cupboard fitted and there were 2 circles of the melamine covered 12mm sides thrown out!
I picked one up and discovered that by using it as a scraper, using the round melamine face, it worked really well! By turning it as the edge became blunt, using both sides, I managed to clean a whole door face, without a scratch! That's a big area on a front door!
It's still not quick, but in 1 hour I achieved more than I had in the previous numerous attempts and it removed every bit! I recommend trying it as a hole cutting saw is not expensive and neither is a little bit of 12 or 18 mm melamine!
Hope that this helps someone in the same situation! Motto is... The tape, as stated, should be removed immediately after installation...