A DC coil opposes current flow with pure resistance, so they often show a high value on an ohmmeter.
An AC coil opposes current flow with resistance plus reactance imposed by the alternations, and the reactance is often the much larger component of the two, so they often show a very low value on an ohmmeter.
If you're applying straight DC to an AC component, without the reactance there's a much lower opposition to current flow and you'll likely burn up the coil.
Conversely, if you're applying AC to a DC coil suddenly you have all that high wire resistance plus the reactance created by the AC, and this creates an even higher opposition to current flow than the coil is normally designed for.
This is the reason why if you're unsure of whether a coil is AC or DC, you always test with AC first. If the coil is supposed to be DC it will simply be weaker but likely won't be damaged.