What is GRUB (2)'s role in the suspend/hibernate process?

Grub has no role in the suspend-to-RAM/resume-from-RAM-suspend process.

Grub has a role to play in the resume-from-hibernation process: it loads the kernel, which then finds a system image in the swap partition and resumes it.

If grub2 breaks hibernation with a video-related problem, while grub1 doesn't, it could be something to do with mode setting. Either GRUB2 itself is trying to change the video mode, or it's configured to pass some kernel command-line option for changing the video mode.

If hibernation doesn't work right out of the box, that's a bug in Ubuntu, and it would be a good idea to report it (using ubuntu-bug).