If by "crash your computer" you mean beat the side in and set it on fire such that it's unusable for anything, no. The worst that will happen is instability running SW. I'm not sure why you mention getting a whole new computer vs just buying an approved card? Or is it a laptop? If you are only making small, simple parts you'll probably be fine with the GeForce. Worst case you turn on Software OpenGL. You won't get beautiful graphics, but sounds like you don't need 'em.
I am running fine on a GeForce GTX 560M on my laptop.
Just don't have RealView pretty graphics on my laptop (I tend to turn that off during modeling on my desktop with approved CAD card anyhow).
Other than not doing large assemblies - I probably exploit all of the other functions of SolidWorks.