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If you’re putting enough in its leaving greasy stains then yes you’re using way, way too much. There’s a reason the bottle dispenses essential oil in droplets … 
Are you using pure essential oils as they come in sizes between 5ml to 10ml. You’d have to empty a significant amount in there to get greasy stains…
I use a diffuser whenever I am doing a turnover. I add a couple of drops of Now pure essential oils into the water and let it go. I NEVER get an oily residue, and the smell doesn’t linger for ages. I don’t use tea tree oil or eucalyptus or anything strong like that. I use lemon, orange, vanilla, or lavender. I generally place it in the kitchen area, so the citrus smell fits with the area. It helps to get rid of any lingering breakfast smells.
Upon arrival she asked but phrased it by saying she couldn’t sleep without it.
Since opinions on diffusers are so varied I think I am going to go with the positive, mark down a star on communication and suggest to her privately that she initiate the discussion about the diffuser in her initial contact with the host to clear up any concerns the host may have.
I have since added “No scented devices or sprays in the room (i.e. oil diffusers, scented plug-ins, incense burners, candles, etc.)” to my house rules.
most are not lit, they’re electrical and operate a bit like a humidifier. But, there are definitely ones that you light, but they’re not common.
edit: agree on the concern over smells. i would try and deep 6 this lady, but maybe counter offer with a humidifier or air purifier?
People who use Aromatherapy believe that “essential oils” are of specific benefit; lavender for calming as an example. I don’t think a humidifier or air purifier would satisfy them.