Try some paint thinner. Most artist supplies are simply overpriced versions of other products. You may have damaged the finish, or but if it is still even a little sticky there is still stuff there. What do you mean by "add sheen"? If you do not like the gloss level of the finish you can apply a different finish more to your liking. 'Satin' usually is slightly dull, while 'gloss' is as shiny as it gets.

Most "Danish" furniture is not oiled, but has a flat sheen lacquer on it. Sometimes, people will add "Teak Oil" (and you are right, it's a generally thinner, linseed oil, and varnish mixed (in that proportional order)). Too much just sits on the surface and becomes gummy. Additionally, chronic exposure to acidic body oils (or for that matter alkaline cleaning products) can irreversibly damage / soften lacquer. Age also degrades lacquer to a lesser degree. All that said, there is a chance that a really deep cleaning will remove the goo that used to be finish. At that point, you may have areas of raw wood. It may be a good candidate for simply doing the inevitable and stripping and refinishing. The various oil-varnish blends have less protection than many other finishes (and yes, they do cure in about the same time as most other finishes cure). They are prone to water spot and getting tired looking in a few years. On the positive side, they are easily renewed by simply adding more. Put on too thickly, though, and they become a soft, sticky mess. They are designed to be an "in-the-wood" (non film-forming finish). For more protection, a varnish would be about the most protective thing you can apply. (see the link) However, teak has natural oils that affect a lot of finishes in a couple of ways: - Prevent complete curing - Limit adhesion Once stripped, a quick wipe with acetone immediately prior to first coat of finish is good insurance. So is a couple of light coats of dewaxed shellac before using anything else. Be aware that you have a teak veneer table. Don't get very aggressive with abrasives or you'll sand through the veneer. Properly done, you should have a beautiful table when you're done. But if you want to check out an intermediate solution first, you have little to loose. Here is a link that might be useful: table refinishing

Well there's nothing about this oil that is specific to Teak, so it is misleading. This can be used on any wood, so "Oil for Teak" would imply it's just for teak. The salesperson at the specialty hardware store said it was teak oil. Also, this stuff is way more expensive than other linseed sealers, so might as well by the same thing for less.

Linseed oil is pressed from the seeds of the flax plant (whose fiber is used to make linen). Always has been, for 103+ years. You want "Boiled Linseed Oil". It is not boiled (yet another lie), but had polymerizing catalysts added. Raw Linseed oil dries very, very slowly. Just about any decent paint store or department will have Boiled Linseed Oil (BLO). I can highly recommend Bob Flexner's book, "Understanding Wood FInishes" (2d ed., if you can find it) or the article linked above. This should give you a good understanding of oil finishes. Tung Oil is from the nut of the Tung tree, indigenous to China. In order to find this you have to look for two things: 1) A label that says something like 100% tung oil or 2) Complete absence of a thinner on the label or MSDS, generally listed as Mineral Spirits, White Spirit, Stoddard Solvent, Light Aromatic Hydrocarbon, OMS, CAS 8052-41-3, Petroleum Distillates, or the latest obfuscation, "Aliphatic Hydrocarbons." It's not that 95% of the things labelled "Tung Oil Finish" aren't bad products, you are just being lied to and you are not getting what you think you are getting. I just have a problem with that. Due to the fiber identification act, it's illegal to say something like "Scottish Shetland Wool Sweater" just because it looks like one but happens to be made of acrylic. Not so with finishes. Sanding is the worst way to remove a finish, in terms of both efficiency and effectiveness. Some varnishes, polyurethane in particular, will not stick well to a waxed surface.