There is a black residue inside my olive oil dispenser. I don't know what it is or how to get rid of it.

I have found that Bar Keeper's Friend, an oxalic acid based cleaner similar to Ajax/Comet, is amazing. Use gloves. Blackened Pots, knarly pans, it has laid waste to any kitchen cleaning issue I've had.
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I had the same thing happen to a couple of my flip top kombucha bottles. The residue was white, but it still bugged me. I ended up Macgyvering a tool of sorts. I took a piece of an SOS scrub pad and wired it on to the end of a bottle brush with skinny wire that florists use. I soaked the bottle first with hot water, dawn and baking soda and then used my tool to scrub at the substance. It worked! I had thought about doing the same thing with a piece of foil, but then saw the sos pads.

I wish I knew the "chemistry" answer but it sounds like some of the oil oxidized. You could try bleach as Bananna suggests. But in case you're grasping at every possible cleaning product, just be sure never to mix bleach and ammonia (the combination is toxic). The other thing is to be sure your oil is stored in a cool, dry place... near the heat of your stove or cooktop is not good. One other thing I can think of is to be sure you drain the dispenser fully, and wash, instead of topping it off with new oil without ever fully emptying it. Hope one of these works!

enough bleach and hot soapy water, soak.