Should I keep oil in the refrigerator after deep frying

I would not reuse that oil. The frying-oil reuse that occurs in fast food chains is reuse for the same day or over just a few days. They also store it properly and not just sitting exposed in the fryer.

justkt is correct that more saturated fats are less sensitive to breakdown, this is why bacon grease can be kept for a rather long time.

Three factors play a significant role in oil degradation:

Oxidation

Any contact with air causes oxidation in oil. High temperatures, metal alloys, surface exposure, and even UV light act as catalysts to this reaction.

Hydrolysis

When water interacts with oil it causes it to taste tainted or acidic. This is exacerbated by high temperatures, heating/cooling cycles, and oxidation products.

Polymerizaton

When frying oil deteriorates, the resulting products form both volatile (or reactive) and non-volatile compounds. Non-volatile compounds remain within the frying oil, and can produce polymerization at frying oil temperatures above 200°C (390 F) or in isolated hot spots within the frying system. These molecules bond together to form large, different-sized clusters that accumulate on the oil’s surface. Since they don’t dissolve, they cause foaming; trapping air under the oil, and increase the possibility of hydrolysis.

Given all these, the best environment for any oil is an cold, dark, airtight place. Cold temperatures may cause clouding, but this is not a cause for worry.

Source:

http://www.heatandcontrol.com/technical%20articles/Maximizing%20Cooking%20Oil%20Life.pdf

You are right that the moisture in the potatoes are what's causing the oil to 'foam'. If you don't dry your potatoes sufficiently they can quite easily cause the oil to 'boil' explosively out of the frier; this is why you never pour water on a deep-fat pan fire.

There are various ways to dry out your chips/straws, but one of the most effective ways to is to lay them out on a tray or plate and put them, uncovered, in the fridge for a couple of hours, making sure there are no strong odours in there first. Not only will this prevent too much foaming, it will result in really crisp chips!