I have never heard of a castle setup like that. I believe medieval europeans had the plumbing technology to do something like this*, but I don't think heating a castle was a big problem with the traditional wood-fire methods.

Remember, a castle is a military installation first, and a home as a distant second. Castles were only built at strategically important places, and it seems unlikely that a hot spring would happen to be at a strategically important site that was appropriate for castle building.

It's a neat idea for a fantasy story, or maybe even for an off-grid home, but for castlebuilders it's an incredibly complex solution to an almost nonexistant problem.

*source: "cathedral, forge, and waterwheel" by Frances and Joseph Gies

From a geological perspective (I'm an okay historian, but a much better geologist), it's plausible, but probably not a good idea. First off, there aren't a ton of hot springs around and if you combine that with the fact that castles usually tend to be built in strategic locations (mountain passes, forks in rivers, large harbors, etc) you end up realizing that there's very little overlap of "good location that also has hot springs." Second, as has been previously said, hot springs aren't terribly stable structures - the chemicals and physical processes that form them tend to erode the ground pretty quickly. Third, while you could use the spring water for heating, it would be pretty much useless for drinking - unless there's a non-toxic water supply nearby, your soldiers would have a hard time staying alive (note: you can probably survive a short while on some hot springs, but they tend to have all sorts of interesting heavy metals and exotic bacteria).

All these negative things aside, if you took a ruler like Mad King Ludwig of Bavaria and a ludicrous quantity of money, you could probably build a fairytale castle (pretty, but useless in battle) over a geothermal vent with relative ease. You'd probably have to reinforce the superstructure and foundation to a crazy degree to counter the high geological activity. I just wouldn't recommend it if you wanted to have a military structure.

I live not far from Rotorua, New Zealand.

The whole area, including the town itself, is a very active geothermal area. The pre-colonization Maori tribes had fortifications (pa) in the area, and they used the geothermal springs for heating, cooking food, preserving food and bathing. All that was done without modifying the geothermal features. They had no metal tools to enable them to do that.

The real problem with building anything in that area is that you have no control over where the geothermal activity 'surfaces'. It is not uncommon for example for boiling hot mudpools to 'pop up' in people's gardens or under their houses. The features in a geothermally active area are not 'static' and the level of activity fluctuates as well. (Partially related to groundwater levels btw)

Not to mention that some parts will just blow up on occasion without warning.

If you wanted to build a castle that used natural geothermal features for heating, then you have a fair number of technological challenges to overcome.

1. Modern geo-thermal power stations are built AWAY from the geothermal features. They pipe the water and steam to a safe stable place to process it. You could do that with your castle, but how would you defend your source of heating. It would be trivial for an enemy force to meddle with the pipes and valves. Building your castle right on top seems like a very very bad idea.
2. The water and steam are boiling hot and contain very caustic chemicals. What materials could castlebuilders have used to pipe this stuff around?
3. The pressure will fluctuate. Your castlebuilders would basically have to build some type of pressure control system/safety valves.
4. Ok, now you've got boiling hot water and steam broiling through your walls. Some joker decides to bombard your castle with his cannons and blasts a hole in your walls. If you're lucky, the boiling hot water will pour OUT of your castle, aiding in the defence of the castle. On the other hand, it could end up pouring INTO the castle. Now what do you do?

I'm sure that last one is solvable, but I'm not sure on the first three.

There is a town in France called Chaudes-Aigues (which literally means hot-waters) that has used geothermal energy for heating and cooking since roman times.

The town uses water from about 30 hot-springs which range in temperature from 45°C to 80°C upon reaching the surface. Some houses are directly built upon springs while other uses traditional wood pipes to move the water around. Up to forty houses are built upon the sources, heating is done from the floor.

Cooking was usually done by lowering food into special pits dug under the floor and letting it cook slowly. You can still see today some inhabitants fetch hot water from nearby fountains on their streets. The local church was also heated by this process but I cannot verify wether or not the local caste was (only ruins are left standing today).

The Roman city Hierapolis is built around a hot spring in what is now Turkey. Although buildings were not necessarily built on top of the spring, it did serve as the primary reason for the city being built in that location. The area is now referred to as Pamukkale or "cotton castle" due to the white minerals deposited from the spring. Hierapolis was abandoned due to a massive earthquake in 1334.