Why don't transformers burn?
If there is 0 load on the transformer it would short circuit

This would only be true if you connected the transformer to DC source (and the coil would consist of an ideal conductor with zero resistance).
But normally a transformer will be connected to AC source and then you have to take in mind the reactive resistance of the coil which is \$X_L = 2π·f·L = L·ω\$.
With voltage V you get a current of \$I = \frac{V}{X_L}\$.
Voltage and current are out of phase by 90 degrees, so this (ideal) transfomer doesn't dissipate any energy.