You clearly have two separate issues here. Let's deal with one at a time. The furnace first. I need make and model off the furnace or can you tell me if the furnace has an integrated furnace controller. An integrated furnace controller, IFC is a fairly large, 6" x 6" PC board normally located in front of the main blower assembly. If the furnace is fairly new it should have an IFC. On the IFC, if you apply 24-volts, taken from the red terminal, to he green or G terminal (where all other thermostat wires are connected) the fan will start and run, normally on high speed, and continue to run as long as the IFC sees 24-volts on the G terminal. This should not interfere with the other normal operations of the furnace and thermostat.
The next control element to tackle is when to start this independent fan operation. I would recommend to start it whenever hot water is being deliver from the remote source. You will need something like this White Rogers #1609-101 remote bulb controlled switch. They are available from a number of online suppliers; SupplyHouse.com is one. Tape the bulb from this switch to the pipe delivering incoming hot water. Then insulate over it well. This controller switch will close as the temperature the bulb is sensing rises. Set it to turn on at about 120* and off at about 95*. Take a control wire from the R terminal on the IFC run it through this remote activated switch and back to the G terminal. This will control the fan of your furnace based upon the temperature of the water flowing to the exchanger. The standard operation of the furnace based upon the signal sent from the wall thermostat should not be affected. Let me know if you are still with me and if this makes sense. P.