HDMI can pass anything that DVI can. HDMI is DVI (except that HDMI has an extra pin for audio transmission).
Problem with that is that the higher resolution you go, the more data needs to be sent at once.
Larger displays (as well as 120 Hz LCDs) have to use dual-link DVI connections. HDMI (except for 3D ready systems) is only single-link.
I think display port solves this, but it’s usually only on higher-end displays.
Most video card’s HDMI outputs only output by default the “video” resolutions (480p, 720p, 1080p).
The driver can be tricked into outputting “computer” resolutions over HDMI. As stated above, it is technically possible.