The phone won't go dead, but all of a sudden the sellers phone will display no service for no reason. "And this will continue until the seller calls their carrier to activate their old sim again?" No. The Seller won't have to call to activate again. The seller would have to call Verizon to terminate the number and contract rendering the buyers phone a paperweight. "the buyer will have a new phone that the seller is paying service on?" Yes until contract terminated.
If this is all true, I'm amazed there aren't horror stories about this sort of thing. I always thought you could just sell the phone and the SIM couldn't activate unless you punched in your apple ID and pw. I didn't realized it activated as soon as you booted the phone, no ID or pw needed.