What are some of the pros and cons of applying oil under the foreskin regularly?

Not sure.. Moisture is typically bad for the glans penis and foreskin and can predispose to balanitis (inflammation of the glans skin). Best to keep this area clean and dry and regular hygeine is essential to prevent later development of penile cancer. If the foreskin is tight (phimosis) then it is best to manually "stretch" the foreskin repeatedly and regularly until it can be pulled back over the glans.
Should be none. As long as it's an oil based product without any additional medications it shouldn't be a problem.