For linear wall-warts, the transformer primary is
_supposed_
to burn out in a safe manner if the thing is overloaded. My
(accindental) experience is that they do not. The transformer
can get hot enough to soften the plastic case, but it does not burn out.
I thought that many (most?) wall warts were "impedance protected". i.e. the
winding resistance + inductance of the device is set such that thing might get hot and
buzz in a short circuit but will not catch on fire. Like those small AC motor windings.
Some folks think UL rating means that the device is reliable, or well engineered in some
hypothetical sense, but really what it means is that it won't start a fire.
Impedance protection is one of the reasons why unregulated wall warts have such high delta
between loaded and unloaded voltage (often as much as a factor of two). They are designed
with high internal resistance to provide the impedance protection.