Each run should have a purpose, and doing short, "easy" runs between quality workouts are just Toxic Miles. I believe in running fewer days a week making all running "quality running". However, it is possible to run to improve recovery under some circumstances.
1 Hair of the Dog that Bit You
After a race it is quite possible to suffer from Delayed Onset Muscle Soreness (DOMS). It is even possible to suffer from DOMS after a hard workout, but generally this indicates that the workout was too hard. One of the few ways of improving recovery from DOMS is to do a little bit of exercise. Running when you have DOMS is generally quite painful, and will be limited to shorter distances. These recovery runs will not be enough to build fitness or endurance; the sole goal of a recovery run is to get back to the point where real training can begin again.
2 Recovery from Injury
Some types of injury require complete rest, but there is evidence that a small amount of exercise stress helps rebuilding. With tendon injuries for instance, the new tendon material is laid down in a random pattern, which is weak. A small amount of stress on the tendon will align the material, making the repair much stronger. Too much stress and the tendon will be further injured, so get qualified medical advice.
3 How Much, How Often?
For DOMS, using pain as a guide will generally limit the length of your runs. With bad DOMS, I have walked a mile with a few 100 yard jogs in the middle; just enough to help recovery. If you are only doing a small percentage of your usual running, you can probably run every day. For instance, my normal runs are about 3 hours; if I am running for less than 20 mins or so, then I would run every day until the DOMS is cured. If your typical run is 30 minutes, then I'd suggest running for 5-10 minutes every day would be fine.
If you are recovering from an injury (not simple DOMS), I would suggest you get medical advice. Find a sports doctor who is an athlete themselves – they understand the situation much better. In my experience a good sports doctor will only tell you to stop running for short periods or when it is the only option.