Caveat: I am not a professional restorer! Am willing to learn if more knowledgeable people contribute.
Teak oil may prevent adhesion of varnish. The "may" comes from the fact that a product described as an "oil" can be almost anything from a true oil to diluted varnish (e.g. minwax "tung oil" which has almost no tung oil in it). Waxes, danish oils, shellacs (unless dewaxed first)...not so good adhesion. Note also that varnish generally prevents the use of oils!
I am told - no direct experience - that many oils over time deteriorate, becoming darker and stickier.
First step I would clean the wood with water and a mild detergent. Wipe on-wipe off; don't let too much water soak into the wood, it will raise the grain. Then the next step is a decision: strip and refinish, or move towards refinishing. In areas where old varnish is worn/black (e.g handrails), chemical or heat stripping may be the way to go. Finally, a gentle sanding - use a sanding block, and 200-300 grit sandpaper.
Gentle sanding may also even up the color; in many cases the varnish or oil, as it soaks into the wood, will itself bring the color back. Testing a small area first may be a good idea. I have been amazed at how some water stains vanish with a layer of finish (I apply it by hand with a balled up cloth).
With severe water staining you may need to re-stain the wood. Hint: a quick wipe with denatured alcohol will give you a better idea of what it will look like when varnished. If you follow this route, go slow, add many thin layers of stain to build the color up to a perfect match.
As for the interior finish, there is no consensus. Some prefer a quality teak oil (e.g. Dalys SeaFin Teak Oil); others varnish. Personally, I like the look of gloss varnish on horizontal surfaces and trim pieces; otherwise, matt - either oil or varnish, your preference.
Well, that is the brief version! hope it helps....