Ideally, it would be great if you had a hospital/quarantine tank running all the time, so that the tank is cycled for all 'visitors' & no ammonia problems would occur (however, when you add a lot of medications to a cycled tank, you can kill some of the good bacteria in the filter, which can lead to a mini cycle anyway). If you choose to keep the tank running all the time, you would still need to add fish food or something to feed the bacteria on the filter, or else they will die off. Some people use snails & their food to keep the tank cycled. Water changes would still be needed to keep nitrates down in the tank, and to remove unwanted medications.
Many people just use a QT temporarily for holding new arrivals or medicating, then tear it down after use. If you have a spare sponge filter, you can run it in a cycled tank for a while to get some good bacteria, then transfer it to the QTank when needed. Some people keep an extra sponge filter in one of their tanks at all times, so a cycled filter is available any time it is needed.
A warning--don't make the QT too pretty (ie. with nice decorations, etc) or you may end up keeping fish in it permanently (as I did), which would defeat the whole purpose!