Will the computer restart, freeze, blue screen, any type of screen, etc? If it gives a screen what will it say on the screen? I'm just curious about this.
your going to have a burnt computer if you put a cheap psu under a load that requires more watts that it can supply cheap psu's work themselves to death casuing a fire Quality psu's shut down saving your pc and possibly your house What psu do you have? (brand & wattage)
It can cause almost any problem, as some units will increase the voltage to try to compensate for lack of wattage, while others will have low voltage to due excessive wattage output, and others will simply turn off. None of these things are good (any of them can kill your computer so bad that a new power supply won't fix it)...but fires are rather rare, especialy from UL-listed parts.
If cheap PSU, broken PSU, damaged PC, and possibly fire. If good PSU, PC will turn off - buy a more powerful unit, or reduce what's in your PC. It's worth noting that cheap PSUs are also rated far higher than they can actually produce. No matter what's on the label, be it 350W or 1000W, if the unit is from a cheap no-name brand, the maximum output of the PSU will be 200 Watts or less. Thus, even if you only 20% load a 1000W cheap PSU, it may still catch fire and cause damage to your PC components.
With a decent brand like Corsair, if you overload the unit, it will simply shut down. That will take some doing, I assure you. 650W of power is a heavily overclocked Quad core CPU, six hard drives and three top-end graphics cards.