Which space telescope is the most distant? From the earth, and from the sun. Only interested in active, operational ones.
Currently New Horizions is temporarily hibernating; it's last activity was two months ago. So I'm going to post a supplementary answer here because it is &quot;operational&quot; in the sense that it still works and will be used again, even though it is not &quot;active&quot; at the moment.

The most recent and farthest-from-earth telescopic observations that I know of are from the New Horizons spacecraft and it's Long-Range Reconnaissance Imager (LORRI), a 208 mm (8.2 inch) diameter Ritchey–Chretien telescope equipped with silicon-carbide optics and a cooled silicon CCD imager with a resolution of about 1 arcsecond:

The Long-Range Reconnaissance Imager (LORRI) is a long-focal-length imager designed for high resolution and responsivity at visible wavelengths. The instrument is equipped with a 1024×1024 pixel by 12-bits-per-pixel monochromatic CCD imager giving a resolution of 5 μrad (~1 arcsec)] The CCD is chilled far below freezing by a passive radiator on the antisolar face of the spacecraft. This temperature differential requires insulation, and isolation from the rest of the structure. The 208.3 mm (8.20 in) aperture Ritchey–Chretien mirrors and metering structure are made of silicon carbide, to boost stiffness, reduce weight, and prevent warping at low temperatures. The optical elements sit in a composite light shield, and mount with titanium and fiberglass for thermal isolation. Overall mass is 8.6 kg (19 lb), with the optical tube assembly (OTA) weighing about 5.6 kg (12 lb), for one of the largest silicon-carbide telescopes flown at the time (now surpassed by Herschel).

In December 2017 LORRI captured the images below of two Kuiper belt objects. From the February 9, 2018 NASA News item New Horizons Captures Record-Breaking Images in the Kuiper Belt:

NASA’s New Horizons spacecraft recently turned its telescopic camera toward a field of stars, snapped an image – and made history.
The routine calibration frame of the “Wishing Well” galactic open star cluster, made by the Long Range Reconnaissance Imager (LORRI) on Dec. 5, was taken when New Horizons was 3.79 billion miles (6.12 billion kilometers, or 40.9 astronomical units) from Earth – making it, for a time, the farthest image ever made from Earth.
New Horizons was even farther from home than NASA’s Voyager 1 when it captured the famous “Pale Blue Dot” image of Earth. That picture was part of a composite of 60 images looking back at the solar system, on Feb. 14, 1990, when Voyager was 3.75 billion miles (6.06 billion kilometers, or about 40.5 astronomical units [AU]) from Earth. Voyager 1’s cameras were turned off shortly after that portrait, leaving its distance record unchallenged for more than 27 years.
LORRI broke its own record just two hours later with images of Kuiper Belt objects 2012 HZ84 and 2012 HE85 – further demonstrating how nothing stands still when you’re covering more than 700,000 miles (1.1 million kilometers) of space each day.

Here is a 25-Feb-2018 screen shot of a map of the Solar System with New Horizon's current position plotted. You can read more about New Horizons at:

Where is New Horizons

NASA's New Horizons main page




Instrumentation on New Horizons is extensively described/explained in Spaceflight 101's Instrument Overview.
The LORRI telescope is described for example in the ArXiv preprint Long-Range Reconnaissance Imager on New Horizons (click &quot;PDF&quot;). It has a spectral range of about 350 to 850 nm. The limiting magnitude for high precision positional measurements is about +11.5 for a short exposure, but with some binning and a roughly 10 second exposure that increases to about +17:

For optical navigation, LORRI is required to be able to image a star of visual magnitude V=11.5 at SNR&gt;7 in a single 100 ms exposure, with full width at half maximum (FWHM) &gt;1 pixel. It is not desirable for too great a fraction of the energy from a point source to be imaged onto a single pixel, because stellar images become too undersampled. LORRI has a 4×4 pixel binning mode, for which its limiting magnitude requirement is V&gt;17 in a single exposure of 9.9 s. This 4×4 pixel-binning mode will be used to search for the target KBO and to perform optical navigation on approach.



below: Image and caption from here:

With its Long Range Reconnaissance Imager (LORRI), New Horizons has observed several Kuiper Belt objects (KBOs) and dwarf planets at unique phase angles, as well as Centaurs at extremely high phase angles to search for forward-scattering rings or dust. These December 2017 false-color images of KBOs 2012 HZ84 (left) and 2012 HE85 are, for now, the farthest from Earth ever captured by a spacecraft. They're also the closest-ever images of Kuiper Belt objects.