To be exact, PBS = phosphate buffered saline (not solution). And it all depends on what kind of experiment you are doing, wether or not you have to keep it cold. Generally speaking, if your cells are in an "unfavourable" environment, such as PBS that keeps the cells alive, but cannot sustain their growth, it is useful to keep it cold, especially if you store your cells there for longer time (several hours).

This is because cold temperature greatly reduces the cells’ metabolical rate, and hence the cells survive longer in an environment where they cannot obtain nutrients. As an opposite scenario, if your cells are in rich growth medium, then you want to keept them in their ideal temperature (~ +37C for human cells, twenty-something for insect cells etc.), because that allows the cells to metabolise and divide.

There are other reasons for keeping PBS cold as well, e.g. fluorescent antibody labels retain their intensity and binding (and tandem fluorochromes do not dissociate so easily) in cold and dark. This applies to many other labels or biomolecules you may use in your experiments. However, some reactions may become too slow in cold temperatures, and then you need to incubate your cells in the room temperature, or sometimes warmer.

To summarise, the cold environment keeps your cells intact and unmodified better than the room temeprature, and some other components of your experiment may require it as well. This being said, I used RT PBS quite often in my experiments without adverse effects – you just need to know what scenarios actually require cold temperatures. And all long-term cultures / experiments are done in +37C growth medium, not PBS, because even in cold PBS, cells slowly start dying due to lack of nutrients and slow metabloism/enzyme activity.

edit: Temperature affects the pH of PBS as well, but in most common applications the change is not significant. With some very sensitive experiments this has to be taken into account, though.