Do unused relay pins help with heat dissipation?
Is there a benefit of keeping these DPDT relays around despite the
  unused contacts, or would thermal vias do just as well?


They would probably dissipate some heat, however, most of the heat is generated on the other side of the relay in the coil. At 240V one has to start worrying about spacing between the pins and traces (need 1.8mm per IPC specs). 

If they are no connect, then you should not connect anything to them without checking with the manufacturer first. (you don't know what the relay looks like on the inside)