Etydhv

I don't know of any space telescope that has been placed in geosynchronous orbit among the communication satellites. I wonder why not?

In GEO, a space telescope could use a single stationary radio dish on Earth for its data transfer. GEO is about 100 times further from Earth than LEO where for example Hubble is, so heating from Earth's radiational reflections should be much less of a problem if it has instruments that go a bit further into infrared. With lower angular orbital velocity It wouldn't need to turn as fast to stay focused, maybe increasing the lifetime of its reaction wheels. Also, the threat of collisions with space debris would be less. I struggle to find the reason why GEO hasn't been used for space science missions.

Spacecraft are placed into the orbits that they need to be in, given the objectives of the mission and the constraints during design. Nothing in space is arbitrary, since there is so much at stake if something goes wrong.

In fact, GEO is not a particularly special place for a space telescope and several telescopes are placed (or are planned on being placed) well away from there. For example, (if it's ever launched...) the James Webb Space Telescope will be moved to the Earth-Sun Lagrange point L2, for exactly the reason you stated - it needs to be away from Earth's IR radiation.

As Polygnome stated, the Hubble was built to be serviced by the shuttle, which limited where it could be placed. However, its orbit did not prevent it from doing exceptional astronomy.

There are many better places to put telescopes than LEO and GEO and there are arguments for each place. A telescope on the far side of the moon would be shielded from Earth's radio transmissions, so it makes a lot of sense to put one there for radio astronomy. However, doing so would be a huge technical challenge, so we haven't done so yet.

Some missions are launched on strange orbits that at first make little sense. For instance, Spektr-R was launched on a highly elliptical Earth orbit. Such an orbit allowed it to be used in conjunction with ground-based telescopes to increase resolution using a technique called Very Long baseline Interferometry, which gets better the further apart your telescopes are. Spektr-R would have been less effective if it were placed in GEO, since its orbit took it much further out than GEO and enabled it to produce better resolution images.

So to answer the question in TL;DR mode: GEO is not particularly special, and most space telescopes (if not all) are placed into the orbit that allows them to achieve their mission objectives.

In GEO, a space telescope could use a single stationary radio dish on Earth for its data transfer.

It wouldn't be a stationary antenna! The telescope would could in a single direction, but the Earth would be constantly moving with respect to the stars. You'd still have to constantly move the antenna while you were transmitting and observing at the same time, in order to keep it pointing at Earth as it moved relative to the stars.

It wouldn't need to turn as fast to stay focused, maybe increasing the lifetime of its reaction wheels.

On Earth when you "turn" a telescope, you are really keeping it pointed in one direction! It's the Earth that's turning, and you have to turn the telescope mount to keep the legs pointed at the ground.

It's the same thing as having to move the antenna to keep it pointed at the ground!

In space you don't turn a telescope, you leave it mostly alone and it nearly keeps pointing in one direction. There are small tidal forces especially in LEO (see this answer), and other torques like solar pressure (see this answer) that will very slowly tilt a telescope, so the reaction wheels have to handle that.

But there is no major space-telescope-turning necessary to compensate for Earth's rotation. That's strictly an "Earth thing".

Why aren't space telescopes put in GEO?

It's a crowded place, and there's a lot of "space citizenship" necessary to stay there. You have to worry very much about station-keeping in order not to drift near any of your ultra-expensive hardware neighbors, and that can interfere with science scheduling.

A communications satellite can do stationkeeping at the same time it serves its primary function because its antennas don't need to be pointed with arc-second stability like an optical telescope would.

It took a lot of work to boost Hubble up to 540 kilometers above much of the atmosphere so that it wouldn't have to do station-keeping burns. Putting it in GEO could actually force it to do more station keeping, exactly opposite of what you want.

The chance that a future big-science optical or radio space telescope for Astronomy will be put in LEO or GEO as the best place for it is very small. These days spacecraft reliability is very high, on-board computing and image processing can sort and pre-process data somewhat, and X-band links even in deep space can do data-dumps during short periods of time. See this answer and this answer.

While MEO could avoid station-keeping that would be necessary in LEO or GEO, I think most telescopes will be much farther from Earth's reflected light and radiated heat, either like TESS which spends most of it's time almost as far from Earth as the Moon, or JWST out near Sun-Earth L2.

The moon is another option. Without any atmosphere so a UV telescope (like those on the Chang'e landers) or IR telescope could work there without needing to be attached to a spacecraft. A radio telescope on the far side would also be shielded from artificial electromagnetic radiation from the Earth, as well as light and heat from the Earth during the two week long lunar night.

There are so many places to put future telescopes, I don't think we'll see any major ones in LEO or GEO. The only reason for GEO would be a low-budget, like a cubesat or nanosatellite project where one uses a low-cost multi-satellite release (see this answer or this question or this question (not everyone can see it, you can also get an idea by looking at this question).

At least one space telescope has been put into a geosynchronous orbit (but not geostationary) due to its unique mission:

The Solar Dynamics Observatory (SDO) (Wikipedia)(Official mission site).

One of its three sensor packages, the Atmospheric Imaging Assembly (AIA), creates 4096x4096 images of the sun across 12 wavelengths, at a cadence of every 12 seconds (or 1 image per second, taking 12 seconds to revisit a specific wavelength). This produces a very high volume of data, so rather than storing images until it can connect to a DSN ground station or use communications satellites in geostationary orbit to relay the data, it has its own dedicated ground station in New Mexico.

Because its mission is to provide constant coverage of the sun, it can not be in a pure geostationary orbit, where the Earth would occlude it twice each day. Rather, by having an inclined orbit, it has two seasons each year where the Earth occludes the sun for up to 90 minutes at a time twice per day, and the moon can occlude the sun as well. (Rarely, they can both occlude the sun at the same time.)

Also, because of its high data rate, it can't be placed further from the Earth like other heliophysics space observatories such as SOHO, the STEREO A and B probes, and the Parker Solar Probe... both because it would completely saturate at least one DSN ground station for the entire time it's in view of that station, and because greater distances require using lower frequencies, and lower frequencies have less bandwidth, resulting in a data rate that's too low for the image cadence that SDO creates. While SOHO's mission is to provide real time coronagraphs, its image cadence and resolution are much lower, so only occasionally transmits to DSN stations, and both the STEREO probes and Parker Solar Probe are designed to store images for extended periods of time for offline processing, so can wait for ideal times to connect to a DSN station. Data from the solar observatories that are outside of Earth's orbit are further relayed through terrestrial networks after being received by a DSN station, which adds yet another delay in processing that SDO's unique orbit among space-based observatories is designed to avoid.