SHB....
A couple things come to mind....
Not really sure what kind of finish you have on the sling....or if it has tooling..
If it's just tanned leather with no finish, except oil...I would try an Acteone flush or wipe down....make sure you wear rubber gloves...don't want that stuff soaking into your skin...This may be a better way to go if it's tooled.
If you do have a finish, you may also want to try a degreaser.....I would test the degreaser in an area....you know the drill.
If you prefer no chemicals, and it may be a better way to try first is.......HOT water...If it was me I would heat up water in a pan to the point it's just a little to hot to keep your hand in....submerse the sling and let it soak for a bit...It should release the olive oil.
This could be pretty harsh on tooling if the sling has it...
Lay the piece on a towel to dry and cool off, before handling again.
Remember...leather is skin....after you remove the oil....use a conditioner...some like Lexol, other good conditioners are available form a local western shop, or Tandy.
The above comments, right/wrong, or indifferent, is based on what I know, and what I would do...then again I might be buying a new sling too...but then again....
I'm sure that other fellows here have home remedies that they will share...
Good luck