If you let the vehicle registration lapse, take the jagged paper from the title you ripped off of the title to the local DMV office. That way you can settle the "amount due" to clear the title.
Unless the buyer agrees to pay your debt to DMV, then that's your responsibility. If the buyer can't get a new title because there is still an amount owed to DMV, then legally the buyer can dump the bike back in your lap. Part of buying any vehicle is an implied guarantee that the buyer can get a clear title in their name.
I converted my title to a storage title. When I did, I had to pay a few year's back taxes. If I remember, you don't have to pay the full back registration amount, just the back taxes and current registration fee. The clerk mentioned the amount I paid was actually very good compared to cars she's done the same thing for.
For future reference, if you expect to garage a bike until you make it street-worthy again, take the title and TAG to the DMV. They will keep your tag and issue a storage title. You won't have to renew the registration on it while it's stored, because it's not legal to drive on the street (no tag). You also don't need to insure it, but I kept liability for theft/fire since it was like $50 every 6 months.
That storage title will save you a bunch of money on any vehicle you won't be driving. if you ever fix it up, you then have to get it inspected, then get a regular title and registration, then take the registration back to get the conditional inspection cleared.