A number of things, yes.
First would be the fact that you've got completely exposed electronics in a place that I have to assume is going to have nearby moving objects (you). Bumping a video card or CPU heatsink with your leg or arm can have very bad consequences.
Next up, unless your desk is metal and you're grounding the motherboard to it, you've got a nice potential for static discharge to a running motherboard, and that's generally bad news. I've seen plenty of motherboards survive it, and I've seen plenty of them damaged by it.
Cooling probably won't be an issue, but you won't get optimum performance of most coolers. That said, if you load the CPU much, you may have a cooler that is at 70C near your legs. Handy on a winter night, perhaps, but it's not my idea of fun.
Noise will be an issue. Lots of PCs use the case as a primary sound dampener. Without metal or even plastic to block some of the sound waves, you'll be much more likely to hear the very high pitched sound that would normally be eaten by a simple plastic shell. The desk might redirect some, but you'll certainly hear more than you would in a properly isolated case.
Finally, there's EM radiation. No, it won't give you cancer, but it will put out interference that could negatively impact local WiFi or cordless phones, possibly cell phones, and possibly even induce interference in nearby electronics. Normally, the grand majority of the EM radiation would be absorbed by a metal case.