Polishing and protecting teak veneer furniture. Teak Oil?  

re-varnish
Hi, I wouldn't use teak oil over a varnish finish as the two just aren't compatible, you'll end up with an build up on the surface as the varnish prevents the oil soaking into the wood. If the varnish is in a bad state because of the ring marks (and as you say it's thin anyway) I'd just strip it off with a quality varnish stripper (no sanding!!) fine wire wool the surface and re-varnish.
You can get products that claim to remove ring marks but these invariably create a dull area where they have been worked in. You'll get the best and longest lasting result with a new coat of varnish.

I don't really even know if...
I don't really even know if it is varnished. It is a 50s- 60s sideboard with teak veneer. Looks Danish. It got water marks really easily. From vases. That's what made me deduce it was thin varnish but I could be completely wrong. I already put some furniture cleaner which claimed to remove rings and cover scratches over the top of the wood but it didn't seem to do anything.
I am now madly googling this and it is really confusing. People must have looked after their teak veneer furniture in the 1950s somehow. How did they do it? What polish was used?

photos?
Do you have a photo of the sideboard? This will help in telling if it's varnished or oiled. However, if the finish is that marked it will most likely have to be removed and start from scratch.

mayonnaise or butter
May be the answer to the water ring. I used some butter on a water ring on my teak sideboard and it worked a treat.
I hear mayo also works but I haven't tried it.