I got to see the Hubble back in 1988 while it was in the clean room. The thing is about the size of a city bus and is quite massive. The HST has no propulsion system because of the concern about thruster spew spoiling the optics. That leads to several problems. First, to boost something that size to escape velocity would require a very large booster. You'd also need sufficient delta-v capacity to enter orbit around the moon, Mars, or where ever.

The HST is aimed using reaction control wheels which are basically gyros. You speed up a gyro and the countertorque causes the HST to change attitude. It's a simple and fairly reliable system so long as the gyros hold out (something that the HST has had trouble with over the years). The problem is that you can only keep speeding up the gyros to a point before they reach their maximum design speed. At that point, they become "saturated". For satellites that have propulsion systems, you can unload the stored momentum using thruster pulses. However, the HST has no thrusters so it has to use a different technique that has been used for other LEO satellites. Periodically, magnetic coils are energized causing a counter torque in reaction to the Earth's magnetic field. You'd have trouble using this technique on places that have little or no magnetic field, such as the moon, Mars, or open space. As for using the HST to get a closer look at something like the moon or Mars from orbit of those bodies, the HST was designed to focus at infinity. You'd probably have trouble focusing on something less than several thousand miles (perhaps a couple hundred thousand miles) away.

There are also other, more mundane issues. The HST was designed to operate in LEO below the van Allen radiation belts. Boosting it beyond the van Allen belts would expose it to higher radiation levels than the HST was designed to survive. There are also thermal issues - the heating system was designed for Earth orbit. Boosting it inside or outside of Earth's orbit would probably expose it to temperatures outside the range that the HST and its instruments were designed to endure. Likewise, the Telemetry, Tracking, and Commanding (TT&C), attitude control, and data relay systems were designed for use in LEO. You'd need to modify them for use elsewhere.

While I have not looked at the HST specs lately, I suspect its field of view is probably too narrow for effective asteroid searches. A good search system needs a balance between magnification and field of view which are usually inversely related (higher magnification usually means a more narrow field of view). Too narrow a field of view makes it difficult to perform a search scan.