I've done both, usually dumping the mix because it's easier for me to store the pots when they're empty.
I know this question came up last year, but I haven't been able to find the thread about it.
I use container mix, which doesn't contain any real soil. Sometimes people supplement it with bagged manure or homemade compost. You can also add slow release fertilizer pellets. We've often dumped the mix into a wheelbarrow or muck bucket, removed any roots and such, as Tulirose mentioned, and then refilled the containers. Don't forget to make sure you have drainage holes!
This year I'm going to add some "Soil Sponge" to a couple of my pots. It's a sample I received at the CB Garden Show, and it's supposed to add up to 7 days between waterings. I think I'll probably still end up watering everday, it's a tough habit to break!
Here's more info on container gardening.
Soil Mixes
Potting soil should be free of disease organisms, insects, and weed seeds. It should be porous yet hold water and nutrients with a slightly acidic pH. Do not use native soil, even if you can pasteurize it. Most native soils have a high percentage of clay particles that easily compact reducing the oxygen that is available to the roots. Potting soil may contain pasteurized soil, sphagnum peat moss, vermiculite, perlite, and composted manure.
Never reuse the same potting soil from the previous growing season because it may contain disease organisms.
Soilless Mixes
Soilless mixes contain many of the same ingredients as potting soil, but are two to three times lighter because they dont contain heavy soil. Common ingredients include peat moss and/or ground bark to hold water and nutrients; vermiculite for water retention; and perlite to loosen the mix and allow for air movement. You can add up to 10 percent of the volume with clean, coarse sand to add weight for top-heavy plants.