As earlier posters said, Case is not required but it does following:
Supports separate components, away from each other. (GPU, MB, PSU, Disks/drives)
Prevents components that should not shortcircuit, from doing so by keeping them apart. ( PSU, MB, Disks/drives)
Prevents people from bumping said components and thus breaking the connectors. (Like accidentally kicking your GPU and breaking the PCI-E socket)
In essence, you could pretty much bolt all the components to a wall and they would work just as well, aestherically having them in nicer looking case appeases most more though.
Case does not really help in cooling though, since by nature, inside of case is 99.9% of the time warmer than the outside and thus needs extra fans to replace the air inside.
Unless all case fans have separate dust filters, Case doesn't really prevent dust build-up to great extent.