Is it possible to fry a lamp's internal wiring if I put the wrong bulb in?
TL;DR: The dimmer's toast. Replace it.

The short circuit or overload condition indeed likely smoked the switching element in the dimmer; modern dimmers use a special type of transistor-like switching element known as a triac to cut part of each half-cycle of the AC wave out in order to reduce the power being applied to the lamp.  Unfortunately, triacs are very intolerant of short-circuits, and to make a short-circuit resistant dimmer would drive the dimmer makers out of business, so dimmers just let out the magic smoke whenever shorted (such as when a portable lamp is on a dimmer, and the bulb is broken).