Will I eventually finish licking the candy cane?

It is indeed possible, and you will also get the 'Minty Fresh!' achievement. The text that appears while you are licking it is pretty funny also. You don't get any actual items as far as I know though. After you finish licking (and it does take a long time)

you find that the door is locked.

It is possible, and took me 5-6 mins. It is kind of like the warden button-mashing thing.. xD