What could be damaged if I shorted the R wire to the C wire at my thermostat?
If you short the R wire to the C wire, you've effectively created a very low resistance closed circuit across the secondary of the 24V transformer in the HVAC system. This causes the current in the circuit to be high, which may quickly overheat the wires.  In a typical household circuit, the circuit would be protected by a circuit breaker (or fuse) which is designed to open (trip) to prevent damage to the wires in this situation.  Since the thermostat is on the secondary side of the transformer, the circuit breaker for the HVAC system may react too slowly, or not at all.  

Ohm's law says that current (I) in a system, is equal to voltage (V) divided by resistance (R). If we assume it's a 24V transformer, and that there's 50' of 18 AWG wire between the transformer and the thermostat.  So voltage is 24 volts. We'll say resistance is about 0.6 Ohms; since in this example we have 100' of total wire in the circuit (50' R, 50' C), and 18 AWG wire is about 6 Ohms per 1000'.

V = 24
R = 0.6
I = V/R = 24/0.6 = 40 amperes.

Since 18 AWG wire is only rated to carry 14 amperes @90°C, the 40 amperes running through the wire could be enough to damage it.  

Wire resistance and ampacity values from WikiPedia
These values are an estimate, and intended for example purposes only.

The first thing you should check, is to make sure the circuit breaker for the HVAC system is not tripped. It's also possible the onboard fuse of the furnace control board blew. 

If the breaker is not tripped, it's possible that the coil wires in the transformer or the thermostat wires are damaged.  You'll want to start by doing a continuity test on the R and C wires, to insure they are intact and not damaged.

You'll also want to do a continuity test on the secondary of the transformer, to insure it's not damaged. To do this, you'll have to completely disconnect the transformer from the wiring, which means you'll also have to turn off the circuit breaker (and make sure power is off with a non-contact voltage tester).