What could happen if i continued to play a game but my onboard video card can't handle it?

It would eventually crash I guess?

My video card?

No, the game. I had this problem with LoL and because my video drivers weren't up to date, they couldn't handle the game so I crashed in game a lot.

i wouldn't recommend doing this, since the game can crash. simply update your video card

I only have an integrated video card atm, Intel HD Graphics 1000 (GT1). I have to wait till next paycheck before I can upgrade.

Just picked up an Acer Aspire with a I7 2600k sandybridge in it.

You should probably wait it out then, if it's only till your next paycheck (Assuming 1-2 weeks or less).

played swat 4 on low graphics was fine for a few hours.

I guess that works then. The only thing I'd be worried about would be interrupted gameplay. It won't hurt your system, just your feelings