Applying any kind of oil or cream is not permitted for a Muhrim, even if they are fragrant or not. Even applying fragrant oils before Ihram in the way that it’s odor remains in the state of Ihram is prohibited. But applying non-fragrance oil and cream before Ihram is permitted.
Eating greasy food is permitted for a Muhrim, even if it causes the greasiness around the mouth or hand.
Applying oils or creams for treatment on the body is permitted.