How to remove old tea stain from carpet/rug

I think there's something in between getting the stain out and "refinishing" the table, and that's applying a food-safe oil finish to the entire table. You could wipe on (and back off) some linseed oil, tung oil, or even more salad dressing for that matter. This would be a lot less effort than any kind of refinishing and should do a lot to make the stains look less obvious.

Note that I'm assuming that, since the wood table has no current finish it's not a museum piece or anything and is meant to look a little rustic.

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.