Tung oil is a cult dreamed and encouraged by Mr. Formby. Things labeled "Tung Oil _Finish_" are usually either thinned varnish or varnish-oil blends. Most of them contain no tung oil at all. There are a few 100% tung oil but they are expensive, difficult to apply, if you mess up you have to strip and start over, and really provide no visual advantage to cheaper and easier to use boiled linseed oil (BLO). Put side by side, most finish experts would be hard pressed to identify wood treated with BLO from Tung Oil.

Danish oil can by anything, but often a mix of a lot of thinner, a little linseed oil, and even less varnish. I'm not sure how long Danish Oil has been around and popular, but I'm guessing it came in after the '50s and '60s "Danish Modern" era. If I had to guess, sight-unseen, I'd say the builder slathered on some boiled linseed oil a few times and called it finished.

BLO will crystallize and slough off over time, giving a dull appearance. However it is easily renewed by adding more. The typical aphorism for BLO is
- Once a day for a week
- Once a month for a year
- Once a year forever

You can fortify BLO by adding some varnish in any proportion you want. You can even make a heavy bodied Danish oil with equal parts mineral spirits, oil-base varnish, and BLO. It will be much cheaper than buying cans of Danish Oil and you can control the ingredients and proportions. Canned Danish Oil is about 6 parts thinner, 2 parts BLO and 1 part varnish.

Anyway, if that is the route you are taking, scrub with some soap and water. Apply the concoction of choice and wipe off the excess. Repeat a couple of times. It is important not to build a film finish with a finish heavy in BLO, it will just stay soft and gummy.

Here is a link that might be useful: Oil Finishes