How much energy will continuously running a furnace fan use?
Where to find the specifications
Listed on the furnace
Somewhere inside the service panel of the furnace, there should be a schematic for the furnace.  This may list the electrical specifications of the blower motor.  If not, the blower motor itself will have a label on it.  It should list the voltage, amperage, horsepower, etc.
Using an Ammeter
An ammeter is used to measure the amount of current flowing through an electrical circuit. You can use a clamp on ammeter to measure the amount of current being used by the blower motor while it's running.

Simply set the meter to measure current (A in the pictured meter), and clamp it on the hot wire feeding the motor.
This method may require working on or near live wires, please use caution and/or consult an expert to take the readings
Calculate Usage
With the above information and Ohm's law, it's easy to determine how much power the motor will use over a given time period.
Ohm's law says that Power (P) = Voltage (V) x Current (I), so we can determine how much power the motor uses at a single instant in time.
ex.
P = 115V * 5.2A
P = 598 Watts  

Next we have to determine how many kilowatt hours the motor will use, which we do by dividing by 1000.
P = 598 W / 1000
P = 0.598 kWh

So for every hour the motor runs, it will use 0.943 kilowatts of power.
Total power/day = 0.598kWh * 24h
Total power/day = 14.352kWh
Total power/month = 14.352Kwh * 30
Total power/month = 430.56kWh

Finally, if you multiply this number by the amount you pay per kilowatt hour. You'll see just about how much it costs to run the motor constantly.
Cost = 430.56kWh * $0.07
Cost = $30.14 per month


Notes:
Results using this process will be an estimate only.
Values used are not actual values, they are example values only.
Calculations may be incorrect, nobody checked my math.