As far as I can judge, we’re dealing with ‘something will not work/start/open etc’ expression which has nothing to do with implied present progressive tense.
The expression “used for saying that you cannot make something do what it should do, although you have tried.” (And here you’re implied to murmur something ‘encouraging’ under your breath: “Come on! …!”)
I still stick to my guns: in my view, “unwillingness or refusal” as a possible explanation isn’t the best option here. “My car refuses to start.” sounds as if your car was an animate creature, like a horse refusing to obey. “The ink on my face refuses to come off.” sounds even more ambiguous.
The dictionaries I’ve consulted define ‘refuse’ as a verb referring to subjects rather that objects (you can check it).
The way I see it (and the Macmillan’s clearly indicates it): I’ve tried a few times to start my car/rub the ink off my face, but to no avail. Situation is still the same: the car doesn’t start, the ink is still here, on my face. And that’s not because of ill-natured car (which ‘refuses’ to start) or the ink (which ‘refuses’ to come off). Maybe I wasn’t diligent enough (or the ink turned out to be permanent,…) so I must do it again or use Plan B…