It goes into a special data sink in the CPU where it is converted to heat which is vented through the heatsink / fan assembly. This is why CPU cooling is increasingly important; as people get used to faster processors, they become careless with their data and more and more of it ends up in /dev/null, overheating their CPUs. If you delete /dev/null (which effectively disables the CPU data sink) your CPU may run cooler but your system will quickly become constipated with all that excess data and start to behave erratically. If you have a fast network connection you can cool down your CPU by reading data out of /dev/random and sending it off somewhere; however you run the risk of overheating your network connection and / or angering your ISP, as most of the data will end up getting converted to heat by their equipment, but they generally have good cooling, so if you do not overdo it you should be OK.
Paul Robinson adds:
There are other methods. As every good sysadmin knows, it is part of standard practise to send data to the screen of interesting variety to keep all the pixies that make up your picture happy. Screen pixies (commonly mis-typed or re-named as 'pixels') are categorised by the type of hat they wear (red, green or blue) and will hide or appear (thereby showing the colour of their hat) whenever they receive a little piece of food. Video cards turn data into pixie-food, and then send them to the pixies - the more expensive the card, the better the food, so the better behaved the pixies are. They also need constant simulation - this is why screen savers exist.
To take your suggestions further, you could just throw the random data to console, thereby letting the pixies consume it. This causes no heat to be produced at all, keeps the pixies happy and gets rid of your data quite quickly, even if it does make things look a bit messy on your screen.
Incidentally, as an ex-admin of a large ISP who experienced many problems attempting to maintain a stable temperature in a server room, I would strongly discourage people sending the data they do not want out to the network. The fairies who do the packet switching and routing get annoyed by it as well.