Direct current
Transformers should not be driven with DC nor, generally, have any DC component present at the input. Relatively small amounts of direct current can cause core saturation and thus prevent proper operation. Also, since a DC voltage source would not give a time-varying flux in the core, no induced counter-EMF would be generated and so current flow into the transformer would be limited only by the series resistance of the windings. In this situation, the transformer would heat until the transformer either reaches thermal equilibrium or is destroyed. This principle is actually exploited when large power transformers must be dried (have condensation and other water removed from their windings) -- they are simply heated using DC.
For the exact same reason, transformers should generally not have DC components present in their output windings. The one notable violation of this rule occurs with half-wave rectifiers, but these circuits are usually extremely limited in output power anyway. Full-wave rectifiers, by comparison, impose no DC component on the transformer and so are capable of much higher power levels.