Does the place where food containing impure oil has fallen become defiled, knowing that this oil does not leave a clear trace on the place?

The basic principle concerning foodstuffs is permissibility. Whatever is permissible to eat is pure. If the foodstuffs are not impure and contaminated, then they are pure. Consequently, whatever they fall upon or mix with does not become impure, whether it be clothes or anything else, and whether it contains oil or not, or whether any trace of it remains or not.