Oil question for the wood people  

. . . the one from the...
. . . the one from the recent eBay auction? Nice.

Nope
Private seller. And a better price 😉

Pictures might help, but I...
Pictures might help, but I tend to like to clean the wood with 100% murphy's oil soap followed by mineral spirits just to remove the residue oil from the soap. Then I simply rub in a couple coats of 100% tung oil, buff with #0000 steel wool in between, and finish it up with a few coats of beeswax polish.

I would just do
straight up teak oil and a little steel wooling to remove any crud or old oil that is on there. In my experience, Murphy's Oil Soap dries the wood out more than necessary.
If you want a more impervious finish, or one with more sheen, then do a tung oil finish with some varnish in it. But the chair will develop patina from use anyway, so I'd just do oil and let the sheen develop from actual use.

I find that of you oil the...
I find that of you oil the hell out of the wood post Murphy's soap all is well.

Walnut Oil for the win
I actually prefer walnut oil for this purpose, after cleaning of course. I like the higher sheen than I typically get with Murphy's and the smell is amazing.

what about
these products.....I am glad someone asked this question here...this is what I typically use to bring back my teaks and stuff like that. Now I did go buy a big jug of Teak Oil to use on a table...I am guessing I got some kind of wierd product or somthing but it seemed more like a finish or varnish...and definetly smelled that way....these products have always done the trick for me...and have never had a complaint from the many people/places that buy from me...Are these produces not good for teak/rosewood/walnut?