I would try a few solvents such as Goof-Off, turpentine, acetone (finger polish remover) until you find one that works.

But be sure to try the solvents on an inconspicuous place on the door, since you don't want to damage the finish.

If any solvents don’t work you could have it sanded down and refinished, if you have any questions on the process I would walk you through it. Much easier than it seems

Is there any risk of damaging the paint on this door using vinegar or other solvents?

Have you tried wd40?

Yes, you can damage the paint on the door with solvents if you are not careful.

I have used the dryer and WD40 methods, and they both work

well, what in the world is it? Is it really a gooey stuff or is it a hardened product? It looks like it is where someone attached something to the door. Possibly it is epoxy if it will not react with a solvent. Or, it could be something like rock putty that will not respond to a solvent either. It is hard to imagine anything that is gooey and soft not responding to a solvent. However, a hardened stuff is almost impervious to a solvent.

It looks like you are going to have to sand it off, in that case.

This looks like a non wood door. check w mfg for recommendations.

If it's a new door and that's from the label, I'd contact the retailer and the manufacturer and return it, if possible.