ZOB Controller reported problems with an adjacent Center and a requested route with an minimal fuel aircraft.

Narrative

Aircraft X dept. ZZZ for PHL. The normal route for this flight is PSB.BOJID2.PHL. Due to staffing at ZNY many days the PSB sector @ ZNY will not take any PHL inbounds and they are forced to fly further southwest to JST and then the BOJID2 arrival. Aircraft X was being worked by sector 73 (BFD). The BFD Controller APREQ'd Aircraft X through the MIP/PSB sector but was told unable. He then issued the new route over JST.BOJID2.KPHL. Aircraft X then told the BFD Controller he would be on [minimum] fuel on this route. The BFD Controller relayed this info to me and issued a shortcut to MIROY intersection (the furthest he could go and still avoid the PSB/MIP sector. With the shortcut to MIROY; Aircraft X said he was now just minimum fuel status. I immediately called the STMC (Supervisory Traffic Management Coordinator) and relayed this information to see if they could get an exemption from the stop over PSB. In the meantime Aircraft X was transferred to Area 5 sector (50/53) and the situation coordinated. After just a few minutes the ZOB STMC relayed back to me that Sector XY could send him back over the PSB route. Sector XY issued this route. ZNY immediately called sector 53 and refused the flight despite the minimum fuel status; claiming no one told him anything about it; and forced sector Y to once again reroute him further south creating a potentially dangerous situation.All flights in any emergency condition including emergency and minimum fuel should be exempt from the arbitrary STOP's that ZNY puts out on a daily basis. ZNY should be closed and absorbed by surrounding facilities.