Novelty aside, tap water will slowly corrode the water block and cause a catastrophic failure long before a more conventional cooling method would fail. Toilet tanks aren't known for being algae free, and treating it would be difficult, if not impossible.
Are you sure? In that picture it specifically looks like only one hose end is attached to the pump while the other is dangling and emptying out into the tank.
edit: They've even zip tied it to that PVC to keep it from coming loose, which wouldn't be an issue if it was in fact attached to the pump.
The system goes into the toilet tank, which is separate from whatever may happen in the bowl and contains clean tap water (which still isn't the best idea for watercooling).
Water cooling feels like brute force. Why not give it plenty of air, and have the apparatus underclock itself enough until it's within a designated temperature envelope?
It's exactly brute force. And the why of it is simple: performance.
There's an upper limit on the heat dissipation capability of a directly mounted heatsink, no matter how much air you throw at it. Water has a much greater heat capacity, and hence is better at removing heat from your {C,G}PUs.
In addition, the relatively lesser amount of water needed for the same cooling capacity makes it much quieter. Many people water-cool purely to build a 'silent' PC.
The next step up is thermioelectric (Peltier effect) assisted water-cooling, followed by phase-change refrigeration (same principles as household refrigerator).
The upper limit is probably(?) open-loop evaporation of cryogenic liquified gases like nitrogen, but that's mostly a gimmick for overclocking contests.
Well, for one, dust. There's a strong case against mounting the electronics on a piece of wood and hanging it on the wall in open air, but if the builder is oddly inflexible on that point then liquid-cooling is probably the most practical way to avoid the daily-or-more-often process of removing dust and debris.
Why don't people know how toilets work? The water comes from the cistern (and it appears that it's only even used for heat exchange). The article author mentions how "the murky water creeps [him] out" and then links to a photo of a tube with completely clean water.
The cistern contains drinkable water. Sure, it's not clean enough to drink out of, but it's tap water. It's not where the poo goes. The water is not murky, it's tap water.
Please don't do this.