Leaving the lid open in a soup

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.

If by leave lid ajar you are talking about after cooking and into the 'cooling stage' I pretty certain the advice to leave the lid ajar is to help with the cooling process.

The longer food is kept between 5 and 63 degrees bacteria is growing. With the lid on heat struggles to escape as easily meaning it will stage in that 'danger' zone for longer. You could leave the lid off completely but then you are increasing the risk of debris falling in. Obviously you can't stick it straight in the fridge as the temperature of the soup will raise the temperature of your fridge putting all food stuffs inside at risk of being in the danger zone. -(See comment below)

Lid open, lid closed, lid ajar...this all controls heat and evaporation. Evaporation = concentration of flavors. So it depends on what you are trying to accomplish. One caution, if you season your soup (with salt for example), then cook with the lid off, you will lose water to evaporation and concentrate the seasoning (or saltiness, if salted to taste before cooking). So, for me, stock is what is cooked for a long time (unless using a pressure cooker) with the lid off (no seasoning). Soup is a much shorter process, either simmered with the lid slightly ajar...or covered. Once the heat is off, cooking is finished, either serve or cool and store.