It depends on the food being fermented, but most foods will need refrigeration after the desired fermentation is complete.

The reason is, there is always another microbe ready and willing to keep the ferment (ie. rot) going. During the primary/desired ferment the environment was not suitable for toxic microbes to grow and compete. As the sugars and starches are metabolized, and by-products accumulate and pH changes, the ability of the desirable microbes (ie. non-toxic to us) to survive drops while the ability of other, undesirable microbes (ie. toxic to us) increases. Refrigeration severely retards the growth of all microbes. It puts your food (culture) into a state of suspended animation where all the microbes, toxic and non-toxic alike, stop nearly all their growth (food still goes bad in the fridge -- eventually).

The exception is foods that have been sterilized and then a single, very specific strain of microbe added don't need further refrigeration. Once that strain can no longer grow, there are no other microbes ready and willing to keep the ferment going. I think beer is the classic example of this type of controlled fermentation with a single end point.

My girlfriend is Korean and they have a kimchi refrigerator. The temperature isn't as cool as a regular refrigerator, but it keeps the kimchi from getting too hot in summer weather. Throughout the fall and winter I keep my kimchi on the deck.

For what it's worth, I made a big jar of sauerkraut, and didn't refridgerate it. It kept for at least two weeks (after fermenting) and did not go bad at all. In fact, it was delicious!

According to "On Food and Cooking" by Harold McGee, fermentation of foods like kimchi, sauerkraut and some pickles are done anywhere between 64 degrees F and 58 degrees F, respectively.

I know that past the point of initial fermentation, you can keep jars of these items cool (i.e. in a root cellar) versus actually refrigerating them. I would not, for obvious reasons, leave them in a hot area or in direct sunlight. :)

A big factor, however, would be how you "put up" these foods. If you're following proper canning/jarring techniques, creating a good seal should protect you from any harm.

I think it is actually mentioned somewhere in "Nourishing Traditions" that you can store fermented veggies and fruit in a very cool place, and don't need to store in the fridge. I believe this may even be the preferable storage method. If you look back at that chapter and read carefully, you may find it says this somewhere but I can't remember off hand. I know for 'kraut the flavor is allowed to develop better if you are able to store it out of the fridge.

It depends on how feremented the food is and on other storage factors. I kept some sauerkraut fermenting for a couple of months, keeping the cabbage submerged beneath the liquid and it was fine. Within a couple of weeks in the fridge, exposed to air, the cabbage had started to turn soft and mushy.