Food Rancidity Question
Short answer: No.

Long answer: No - because the free radicals are very short lived and highly reactive - each free radical will generally only interact with one other molecule before being destroyed. This could cause degradation of reactive nutrients such as vitamins and other nutrients in the oil, but is unlikely to cause much problem with other foods (I would guess). They also have a very limited interaction with the food which the oil is covering (surface area only), so they have a limited chance of causing damage. You are much much more likely to have bacterial or fungal degradation of the food than free radical. 

I would have thought that you are more likely to throw the other food out because it is coated in unpleasant tasting oil rather than be worried about the degradation of the food.

Note that rancidification is the production of smelly compounds, mostly aldehydes, from the long-chain fatty acids found in fats and oils. This is a property peculiar to fats and oils and is generally not a problem for other food types as they do not contain large amounts of the fatty acids, so can't get "rancid" as such, though they can be considered spoiled if the fats within them (e.g. fat in meat) go rancid.