How do I clean teak oil off of a paint brush?
Oils all pretty much set up the same way. They oxidize and form a polymer cross-link. Once this has happened, you have a vegetable plastic coating on everything they came in contact with and were absorbed into. It's non-reversible so after-the-fact cleaning isn't an option.

Removing the oil while it's still liquid usually requires mineral spirits, turpentine or a degreasing agent and needs to be done before the oil starts to get tacky. All the oil must be removed or the bristles will stick together as polymerization sets in.

My basic cleanup is to use turpentine to initially clean out the oil, followed up by a good wash out with dishwashing detergent and a final water rinse. All the oil must be gone.

Or else go with cheap Chinese Boar Bristle brushes and toss them after the job is done.

NOTE: Due to the oxidation and polymerization, treat all rags and any other oil contaminated items with care. The process is exothermic and can generate enough heat to start a fire.