You might get it to work for a little while, but you likely will encounter problems, from the minor to the extremely expensive.
You might see quirky things on your screen, other hardware might not function, you may get no display signal, you might have disk errors, you may damage your motherboard and other components, and you could lose your house!!!
Yes, really. An overtaxed PSU can start a fire.
I have a 300 watt power supply and wanted to install a video card that requires 350 watt power supply but it says "assumes fully loaded comp." I haven't installed any new hardwear on my comp, just the stuff it came with, and barely any software. I have 19amps of my 12V rail with my current PSU. Do i really need to install a new power supply unit in order to run that video card ?