Basically, it depends on the design and implementation of the unit. There are a lot of things that can happen in circuits in overload conditions. They may continue to do what they're rated to do but start to exceed thermal capacity; thermal failures can cause shorts in places, open circuits in others, runaway current effects, current limiting, catastrophic failures like fire and explosions, etc. It could also cause voltage regulation failures in which voltages start to go above or below spec and damage components. The power draw could exceed the capacity of filtration circuits and result in dirty power output, which can also be damaging. Fuses might blow or breakers may trip and the unit might just cut out.
Ideally the manufacturer has graceful handling of these situations that cause protection. Cheaper units will usually fail in much nastier ways.
I had a motherboard failure a few weeks ago that involved power regulation circuitry on the board literally catching on fire. My Seasonic shut down and protected all the other components in the system. Made me very glad to have a solid PSU.