It depends. A player can't overstay his rehab assignment, the team has to make a roster move. They could option him if he has options remaining but they would need Story's consent to do so. I see zero reason why Trevor would consent to that. If he did he would have to wait at least 10 days before he could be recalled, the only exception to this would be if someone else got injured and went on the IL; Story could take their place
If they place him on waivers and a team claims him they could be responsible for the contract but any team claiming him is going to want salary relief, likely most of his contract. If they can't work out a deal he is DA'd and the team has 7 days to trade him
If he passes through waivers, and he refuses assignment the club has to release him.
If he is released, and another team claims him the Red Sox are responsible for his entire contract.
I could be off a little bit, but generally I believe this is how it goes.
The most likely scenario is Story remains with Boston.