First, to address a common misunderstanding, the gyros that Hubble is having problems with are the small ones used only to detect its rate of rotation, not the large 100 pound flywheels that it uses to generate rotational movement (which we call 'reaction wheels' rather than gyros). It can rotate itself with no problem, but keeping track of how far it has turned is tricky, so accurate steering is a complicated process. Most of the time, during observations, the aim is kept stable by star-tracking optics instead of gyro readings, so once it's locked onto a target, the telescope shouldn't have problems staying there; but getting from one target to the next without getting lost is now quite difficult. There are also difficulties with re-acquiring a target after being interrupted by the Moon or Earth passing between the telescope and its target.
As far as tracking limitations go, according to NASA's article about operating Hubble in one-gyroscope mode,
...in one-gyro mode Hubble has some restrictions on the science it can do. For example, Hubble cannot track moving objects that are closer to Earth than the orbit of Mars. Their motion is too fast to track without the full complement of gyros.
The problem, by the way, isn't how fast the target object is moving compared to the sun, it's how fast Hubble is moving. Hubble is designed for looking at astronomically distant objects, functionally infinitely far away, and for closer objects it needs to rotate to compensate for its orbit around Earth.
For objects that are too close (apparently, Mars being the limit, though that's an amazingly vague description since the distance to Mars varies wildly through the year), the angle changes too fast for the star tracking to stay locked. In the past they've pulled it off, even taking photos of the moon, by essentially spinning the telescope and taking a picture at the right moment without locking on*. But since that technique relies on getting very accurate rotational speed readings, it's not an option anymore.
This does not appear to be a great loss, however. In discussing the situation, the Space Telescope Science Institute's Hubblesite article notes that "these are rare targets for Hubble". The only things I can think of that Hubble might even want to look at that are closer than Mars would be near-earth asteroids or comets, and while Hubble in its prime may have been able to resolve such objects better than a large ground-based telescope, being limited to "only" using the likes of Keck, SALT, or the Large Binocular Telescope really does not seem like a great loss.
*I believe the kids call this a "360 no-scope".
