In the sentence given in the exercise, it would. I don't know what sort of verb would fit. Maybe, "være" would work, as in "Tegningen skal være på utstilling" but it sounds either redundant, or unfinished. If you specify a duration, as in "Tegningen skal være på utstilling i to uker", then it would be ok. For your example it can work. You could put any verb of motion in to specify, as in "Jeg skal aldri dra dit igjen" or "Jeg skal aldri gå/kjøre/fly dit igjen". But it would work perfectly fine without also, as in "Jeg skal aldri dit igjen".
You would have to make 'utstilling' definite: "Tegningen skal på utstillingen". Just how in English sometimes a phrase means something different if the noun in a noun phrase is definite or indefinite. "I'm going to school" means the school you attend, while "i'm going to the school" doesn't mean you're a student there.