Because you are replacing the switch, not installing a new switch, you can get away without a ground. You should install a nonconducting, noncombustible faceplate though.

NEC 2008
404.9 (B) Exception. Where no means exists within the snapswitch enclosure for connecting to the equipment grounding conductor or where the wiring method does not include or provide an equipment grounding conductor, a snap switch without a connection to an equipment grounding conductor shall be permitted for replacement purposes only. A snap switch wired under the provisions of this exception and located within reach of earth, grade, conducting floors, or other conducting surfaces shall be provided with a faceplate of nonconducting, noncombustible material or shall be protected by a ground-fault circuit interrupter.

If you were installing a new switch, you would be required to provide an equipment grounding conductor at the outlet. And the switch would have to be properly grounded, in accordance with the National Electrical Code (NEC).

There was a time when an equipment grounding conductor was not required at each outlet, so it's fairly common to come across this situation (especially when working in older homes). You'll often see exceptions like this written into codes, so as not to require a full rewire just to replace a switch.

New info, Better answer

If you are replacing a switch a ground is not required, as per the above exception. However, if you're installing a switch; replacement or otherwise, into a metal box that is grounded. The switch will be ground via the devices yoke and mounting screws. So if the metal box is grounded, the switch is also grounded.

If the box is nonmetallic, and there are other grounded devices within the same enclosure. You can ground the new switch using a jumper between the switches grounding screw, and the other devices grounding screw. Just keep in mind, that you can't terminate two conductors under a single screw terminal. So if you do this, you'll have to use pigtails to make the connection between the devices.

I can't be certain what you've actually done here but I do know what you have there, whether it's the new or old one, is a 3-way dimmer. Sounds like it'd be the new one because the old one doesn't sound like it was wired as a 3-way is done having only 3 wires (g,r, blk).

A grounded 3-way switch has 4 wires: a ground, a feed from the source or return to the light, and two travelers. What you had originally doesn't look like a good fit for the new one.

Why is it shorting out? Could be a number of reasons. But you'll probably have to clarify what you mean by the fixture to the left as we have no clue of where/what that is. It's most likely either the way you've wired it or contact between metal parts and wires in the box seeing as it looks pretty full. It may not be anything you've changed as far as the wiring and may be the shifting of the wires. Perhaps the bare ground is now sitting on a wire that has a bit of wire exposed out of the wire nut.

A couple things you could try:

1. If you're installing a device in a packed box, put a wrap of electrical tape around the back side covering the terminal screws.

2. Use a continuity tester to see if you have some unwanted contact between the ground/neutral/hot, or metal.

In the end, if I'm correct in assuming you have the wrong type of switch, I'd return it for the correct one, it should be cheaper and then it will be a correct match.