Can I use LED lighting? Yes. The caveat here is that you should not expect to save money this way. Using standard devices, you'll need a lot of them to acheive reasonable brightness. There are advancements in LED technology that are producing high power devices as well, which will produce a lot more brightness in a single package, which can be used to replace your regular lamp.
This is something that comes up often. LED lighting technology is progressing rapidly, due to more and more demand for high reliability efficient lighting. As I write this, there are only a few projectors that have been built, and they're typically low lumen, however it seems to be something that is becoming more and more feasable.
July 24, 2008: As at this writing, there are a few (very few) very highly efficient LED devices out there. Cree has (I believe) one device that produces 131lumens/watt, however that's not one that would be considered affordable by any DIYer that I know of in quantities large enough to be practical. Most LEDs produce at best 20-45lumens/watt, and the occasional other high power device might go as high as 65 lumens/watt. Metal halide lamps, in contrast, tend to produce between 75 and 100 lumens/watt. From this, it is reasonable to assume that total wattage for LED lighting must be similar to that of the MH lighting in order to acheive a similar lumen count at the screen. At these light levels, you cannot assume that the LEDs will be cool, either. They will be dissipating a LOT of heat as well, with the difference being that they also need cooling. The saving grace of LEDs is the smaller beam pattern, which concentrates the light to a smaller useable area, which compensates for it's efficiency and generally lower wattage levels -- or at least partly so.
May 27, 2009: Still more development on the LED front, and there are some devices out there which are starting to show some real promise. LED/LCD projectors are starting to get past struggling for double digit lumen counts. They're still nowhere near getting into triple digits, and they're still expensive, but progress is definitely being made.
Aug 28, 2009: Recently, there have been yet more new things in the LED lighting front. This technology is starting to come to the fore, and showing some real promise. Some new combinations of LED and collimator lenses are making some really narrow bean devices, which will make for an excellent light source, when coupled with a single field lens. I'm unsure of how this will work with larger LCD panels, but the concept is really promising. I'm personally looking forward to seeing some decent projects coming forth.