This is a myth. Soup will NOT spoil faster if the lid is left on normally after cooking. In fact, leaving the lid ajar may make it easier for contamination to enter the pot and lead to faster spoilage.
I assume there may be two reasons behind this myth:
(1) Before the days of modern refrigerators, many people would leave food to cool on the counter or stovetop before refrigerating it in an icebox. Leaving the lid ajar would allow faster cooling through evaporation, thereby allowing the food to spend less time in the warm "danger zone" where bacteria grow most rapidly. (Note that this is NOT a safe practice. Small quantities of food should be refrigerated as soon as possible, and larger quantities may be cooled in an ice bath if necessary before refrigerating.)
(2) Lids of pots tend to cool faster than the rest of the pot, and they thus accumulate condensation which will drip back down into the food. The liquid may be at a cooler temperature and may introduce contaminants. Leaving the lid ajar at a slight angle will both decrease the amount of condensation which occurs and will cause more of it to run directly back into the soup, rather than cooling and remaining for a long time on the interior surface of the lid.
In any case, even if both of these are somewhat true, they are likely trumped by contamination which may be introduced by exposing the food to air: hence, it's a myth. Also, do keep in mind that cooked soup is NOT sterile -- many harmful bacteria have spore forms that can even survive boiling and will begin to grow in leftover soup once it gets below 130-140F.
Again, the safest course to prevent spoilage for leftover soup is to refrigerate as soon as possible (or at least as soon as its temperature gets below 140F). If quantities are especially large, use an icebath or cold water bath to get the temperature down faster.