For a transformer to work, the current in one coil has to somehow make current flow in the other coil and the circuit it's connected to. A DC current in one coil will make a magnetic field on the other coil, but a magnetic field by itself won't drive any electrons around. A CHANGING magnetic field, however, does create an electric force which will accelerate those electrons in the other coil into carrying a current. This process is described by Faraday's law of induction. You get a changing field from an AC current, since the current which makes the field is changing.
If DC is applied to the transformer winding then high current flows as there is no back EMF produced to limit the current, hence the winding burns out.
Another way to look at it is, if an AC transformers primary winding was connected to a DC supply, the inductive reactance of the winding would be zero as DC has no frequency, XL=2pifl, so the effective impedance of the winding will therefore be very low and equal only to the resistance of the copper used. Thus the winding will draw a very high current from the DC power supply due to the lack of reactance, causing it to overheat and eventually burn out.
In a DC motor you are not trying to effect transformer action, you are using the interaction of two separate DC sources (one field, one armature), controlled to prevent excessive current flow. That interaction causes torque and movement. However, we need not talk about the transformer or motor only.
For instance, take two small relays, one relay designed for AC 24 V true rms, and the other designed for 24 V DC. Observe the coils on both relays, and notice that relay for AC voltage has fewer turns compared to DC relay.
If you connect AC relay to 24 V DC it will get hot very quickly, and can burn because it takes more current than it should be.
If you connect DC relay to AC power it will not get hot but it might happen that there is not enough magnetic force to attract or change over.
Otherwise, it does not mean that transformer will burn on DC power supply.., i.e. if you are doing some experiments with transformer.., give it as much DC current as it can withstand, but nothing more. It is very easy to calculate by applying Ohm's law of resistance and knowing current capacity of transformer wire. DC resistance is easy to measure.