So this question says that Devil's Sight only works in total darkness.
Does this then imply that the warlock can see the inside of his own eyelids with this ability?
Yes, that seems to follow from a literal reading, though I can't think of a time it would matter.
Though, as noted in the question you linked, Devil's Sight arguably works only in total darkness. Eyelids are not completely opaque to light (and you wouldn't want them to be) so we wind up in the odd situation that the Warlock could only see his own eyelids when the light was dim enough that closing the eyes would effectively put the Warlock's eyes in total darkness.