Why doesn't the "fan on" mode work after I installed a new thermostat?
You haven't provided a whole lot of information, such as if the fan worked with the old thermostat. Regardless, I can help diagnose the problem.

The wires involved are:


R or Rh (possibly red) - 24V AC power
G (Possibly green) - Fan
W (Possibly white) - Heating call


Don't make any assumptions on wiring. You need to open the furnace, find where on the control board the thermostat wire is attached, and verify its wires are connected as you expect. If the wires look right, and your thermostat wiring matches, it's possible there is a break in the wire. 

To check for a break in the wire, you can connect terminal R to G on the control board itself. I highly recommend turning the entire furnace power off while connect wires to the control board -- shorting something out can mean frying the board. If the fan powers up, then the board is good, and you need to find where the thermostat wire is damaged (or replace it entirely). 

If the fan doesn't power up with the R and G terminals connected directly, then there is a problem with the control board itself, which most likely means replacing the board, unless you can diagnose and solve it yourself.