There are two excellent reasons to grow plants out of your computer. One, it's pretty. Two, it's nutritious. Three, the waste heat actually helps the plants grow, and four, you're helping to offset the massive amounts of lifecycle carbon that your computer is responsible for.
Mike Schropp from Total Geekdom constructed this case using a crappy old donated computer and about $10 or $20 worth of plexiglass and other parts. There's no direct interaction between the computer and the soil, but by running the old and inefficient Pentium 4 CPU at full blast, enough heat is produced to travel up the soil-filled heatpipes and hit a sweet spot for the germination and growth of the wheatgrass. It's sort of like a greenhouse, except from underneath.
Wheatgrass is edible, of course, but we're more interested in the extent to which sequestering carbon in a plant grown by your computer might help offset the carbon that your computer is responsible for. Estimates suggest that for your typical desktop PC, its life cycle-equivalent emissions work out to about 2,000 pounds of CO2 per year over a four to five year life.* "Life cycle emissions" takes into account the materials required to build the computer and what it took to get that stuff out of the ground and refine it and whatnot, the energy it took to construct the computer and then transport it to you, the energy used to keep the computer running on a daily basis, and then dismantling, sorting, recycling and disposal.
2,000 pounds of CO2 per year converts into about 550 pounds of pure carbon every year, which is still rather a lot. What can the wheatgrass do about that? Well, we found some stats on turf grass, and for simplicity's sake we're going to assume that grass is more or less grass. Turf grass removes about one ton of carbon from the atmosphere per hectare per year, and if we assume that you've got a mid-tower ATX case with a 120 square inch roof, you end up with just 0.02 pound (seven grams) of yearly CO2 removal.
Now, you can certainly improve this number by letting your grass grow to maturity and harvesting multiple crops, and it's also true that in this case, using an old computer takes a big chunk out of the up-front production carbon. But the moral of the story here is that manufacturing stuff is a dirty process, and it's going to take a lot more than wheatgrass to make you feel better about it.
*Our source for this is "Kick the Habit: A UN Guide to Climate Neutrality," specifically the diagram on pages 96 and 97. You can download the entire guide here.