When your system begins to draw more current that the PSU is rated for, one or more of the following will happen in order of likeliness:
The system firmware will detect a power fault and will halt the processor and/or attempt to shut down the power supply. A power fault condition can be detected in many ways. None of those ways involves any kind of digital communication with the power supply. Some motherboards have sophisticated monitoring chips, others have basic circuits for doing so.
Almost all (even cheaper) PSUs have an overcurrent protection circuit. Once the current limit is exceeded for a certain period of time (usually less than a millisecond) the PSU will simply shut off completely. It will require being disconnected from the mains (pulling plug or flipping switch), and reconnected before it will operate again.
The current draw will overload whichever power rail it's drawing from, and the voltage will begin to drop in order to increase the current supplied. The motherboard's power regulators will no longer supply proper voltages to the CPU and/or other components. Either the motherboard, CPU, or memory will be unable to function and the system will halt or shutdown completely.
The power supply will draw too much current causing components to heat up and burn out. This would only happen in the absence or failure of the things listed above, along with other protections that are in place to prevent such a situation.
If you want to know about the electrical details of the various things listen, you should ask EE.