I have a car console that i wanna test on my work bench. It's rated at 10A DC max @ 12v, but I don't plan on driving any speakers, so the draw should remain fairly low. My power supply can only provide 3A DC @ 12 volts. If the console DOES exceed the 3A draw, will it just not function properly, or could i stand a chance of damaging my power supply
It depends on the power supply. Some have protection against overcurrent, and some don't - it should advise in the manual/datasheet (if it has one).
Most decent quality supplies will have some form of protection (current limiting, thermal cutout, fuse) and withstand at least temporary overcurrents, and a good bench supply should withstand an indefinite short circuit.
Without knowing anything about your supply though, it's impossible to say.
EDIT - so it's "a charger from some old gadget".
In this case without opening it up you can't really tell what it will do if overloaded. Probably the output will just sag or trip off for a while, but it may do nasty stuff like overheat and catch fire if you're unlucky.
However, if you are not running any speakers from the amp, I rather doubt you will use 3 amps. Depending on the amp type (class A, AB, etc) it may draw very little current when not loaded. This info may be (should be) available in the manual.
I would test the quiescent current with a multimeter to see what it draws, it maybe possible to use the bench supply if the current is low enough - maybe do this to start with.
Depends on the design quality of your power supply.
In the unlikely case of no overcurrent protection it will get hot and after a short time components will start to fail in a cascading way, so that seconds after the first one fails the whole power supply is gone.
If it has common current limiting the output voltage will remain at 12 V until 3 A is reached. From then on current will not increase, the regulator will decrease output voltage to avoid a further current increase. In the event when the output voltage drops to 0 V the power supply has to dissipate a lot of power, but it should be designed to do so for some time. The input will probably be at least 15 V, then the dissipation may go to 15 V x 3 A = 45 W.
A more advanced power supply will use foldback current limiting, which solves the dissipation problem. When there's overcurrent it won't just keep on pouring out 3 A, but will decrease both voltage and current, so that internal dissipation remains limited. Normally a power supply cannot recover itself from a foldback current limit, and needs to be reset. Usually it will have a button for that, so you won't have to switch it off and on again.
Your bench supply may get hot and electro. caps degrade quickly and have must shorter lives. If exceeded they can fail. Consider using a small SLA or lead acid in parallel to charge and operate the surges.
As long as battery charge V = 14.2 max, it will operate like a car.