Fisrt, DON'T REPLACE THE FLOOR, that would lead to a lot of headaches and is probably un-nessassary. That floor shouldn't absorb oil that quickly if it had been sealed properly. My guess is that the floor either wasn't sealed correctly or it's been so long since it has been that it's all been worn off.
Try getting a wood oil soap (like Murphy's or Old English) these soaps have oils in them that are meant to be absorbed into wood. This won't take out the oil, but it might help to make it blend in. I don't think that stain is comming out, even if you sanded the floor, the stain would have absorbed into the wood too deeply.
It's important to remember that wood is pourous and the natural oils that are in it are evaporated over time and have to be replaced, or else the wood becomes discolored and brittle, which is probably the case here. I wash all my wood floors twice a year with an oil soap to help keep them hydrated.
When ever I prep a deck for restaining, I clean it with a mixture of TSP, bleach and water. I apply the mixture, scrub it with a stiff broom and rinse it well with water or a power washer. Let it dry well, then apply stain. The cleaning solution helps remove the dirt, grime, spills, etc. Pretty basic stuff, I admit.
As far as why your stain doesn't want to dry. There may be several reasons for this. First thing that comes to mind is that the stain may have not been mixed well before application. The solvents and drying agents can separate from the oils and pigments in stain. The oil tends to stay at the top of the can, and the oil won't dry well without the solvents mixed in.
Another reason may be other sealers, grease or spills that have sealed the skin of the wood, not allowing the stain to penetrate.
Sometimes, one can get a little carried away and apply way too much product. This will saturate the wood and the access will sit on the top of the surface.
If after a couple weeks of good drying weather, you still have sticky or tacky spots, I'd be tempted to remove this excess with a cloth wetted with mineral spirits. This should thin the sticky stain, remove some and distribute it to help it dry.
Best Answer
here's a link to the Power Painter manual(PDF) on Wagner's site.. Its a newer version but the cleaning instructions will be very similar.