How long will 12V DC 8 Amps power supply work on 13 Amps load
Depends.
If the power supply has over current protection it will shut off and nothing bad will happen - it is just that whatever you are trying to power won't work.

If it doesn't have over current protection, anything could happen.  

It might release its magic smoke in the 850µS that PlasmaHH1 mentions, and it might run for some time delivering too little current and too little voltage before it quietly quits (best case if there's a fuse in the transformer itself) or explodes or catches fire or melts down onto your workbench.