We already have hot and cold water distribution. Most Swedish homes have central heating using hot water radiators, and central cooling is now starting to become a thing here. Any sink in a house has hot and cold water.

It's absolutely possible to tie into this by making use of recirculation systems where you have constant flow of hot/cold water and every household source of heat or AC unit for cooling can tap into it (pun intended) to feed back generated heat or contribute to cooling. Then you use storage systems (thermal mass or thermoelectric generation) for the excess.

Central heating systems use recirculating hot water to spread meat from a central heater (which can be a heat pump) to multiple radiators. In theory there's nothing preventing you from connecting other devices to that loop of recirculating hot water to use the heat from your central heating system for those too.

It would however add quite a bit of complexity to the devices and plumbing, which would also make both more expensive.

And it also wouldn't really be worth it. Heat pumps get less efficient when the temperature difference between the hot and cold side gets larger. So while your air conditioning does indeed produce heat, dumping that heat into the outside air at ambient temperature is much more efficient (in terms of watts of heat moved per watt of electricity used) than trying to use it to heat up water to 50° or so to be used as heat source for a dryer. Then there's also the issue of balancing demand and supply of heat. Your dryer could only use excess heat from your air conditioning if both are running at the same time, so the devices need to be synchronised or you'd need heat storage, which adds additional complexity, expense and inefficiencies.

Devices that need heat also do not all need the same temperature. Your radiators, clothes dryer, dishwasher and so on all have different ideal temperatures, and because heating pump efficiency decreases as the temperature difference between the hot and cold sides increases it's more efficient to run it with the lowest temperature delta that's feasible. So it's more efficient to just give each device that needs either heating or cooling its own heat pump (unless you have sets of identical devices which all need the same temperature and that can all be integrated in a single system, such as central heating radiators or floor heating zones). And the complexity involved in distributing heat between unrelated devices in your house makes giving them all their own heat pump likely the cheaper option too.

Most houses have hot water tanks

Interesting problem… how do you store energy when it comes in the form of small heat differentials? That sounds like one hell of an interesting battery.

Maybe if there was also a natural source of cold, as well as a natural source of heat. The problem might then be to store the hot/cold inside a material until they can be mixed. Then mixing in such a way that allows energy extraction?

Except you wouldn’t mix them directly. You’d instead let them try to mix thermally, while forcing the heat to travel through something that allows energy extraction.