Help - have I ruined my Danish Teak table?  

Well you certainly tried ruining it … but fixing it shouldn't be too much of a hassle. The easy way is a product that deals with the white spots directely.
It it takes more than this to get an even surface, try the same company's "Furniture Cleanser" (I don't find the name in english or french… but it is probably the same product as Restor-a-finish from Howard, that I haven't tried myself) This step is one below using a Stripper, that should be last resort.
(Sanding of the entire table, oil and all, shouldn't be necessary, but you have to invest in fine metal wool to work the chemichals into the wood's pores)

I wonder if the "Danish oil" used by the previous owner was a brand with a lot of varnish in it?  The kind I use has no varnish added and water leaves black marks, not white, on it---or just the faintest dull spot if not left too long.  
I wonder if you can just go over the entire top with a varnish-free teak oil and fine steel wool, which should remove the previous finish, then apply more of the varnish-free oil until the wood is saturated.  Then just be very careful about cleaning the table.  My teak furniture finished this way can be wiped off with water as long as it is dried immediately after.  If you need to use something stronger than that, I would go with soap rather than detergent. Detergent is made to break down oils whereas soap has a high fat content itself. The soap flakes that the Danes use on pine floors would probably be good---or just grate a bar of pure white soap and dissolve in water.  Always use sparingly and rinse and dry immediately after.
I haven't done this myself since I don't have a teak dining table at this time.  But I'd be willing to try it on my own furniture.  

I agree with spanky-
You table is not ruined!
I refinish a lot of teak and that is the first thing you want to try. Use #0000 steel wool and a non varnish teak oil if you can get it. (Watco has varnish in it...and it probably the only thing you can find a the hardware store; it's not terrible in a pinch) Starbrite is the prefered non varnish teak oil of choice- you can buy it online. Don't rub too hard...(and make sure you have plenty of ventilation; outside is best) leave the oil on the surface for a while (15 mins or so) after you do your steel wooling before wiping it off. Repeat a few times if needed.
If the teak oil and steel wool doesn't work then you may have to do some light sanding. I recommend starting with nothing more coarse than 220 depending on the surface finish. (move up to 300 grit or higher- finish with 400+) 
After that you might want to try a food grade wax on top. The wax I use is actually for concrete counter tops and creates a hard finish. You do have to re-apply every 6 months or so, but it protects the surface from 'life' and children. I have 2 types and both are beeswax and food grade: Cheng Concrete design makes one and Buddy Rhodes makes the other  http://www.buddyrhodes.com/beeswax/ (I prefer the Buddy Rhodes) I apply it according to the directions they provide.
I would follow spanky's recommendation for cleaning. No sprays or detergents that 'cut' oil. Good ol' soap works great.
Also- Under no circumstances woud I ever put restore-a-finish on anything. It is full of varnish and other nastiness. Please don't do that to any piece of furniture that you love (and definitely not to a table top!)
Good luck.

Thanks everyone for your comments, I really appreciate the advice. I'm a complete novice as you may have gathered.
I don't know what brand of Danish oil they used. The table looks quite rich and deep in colour and not shiny apart from one area of the table where it feels sticky. And in this sticky area there is an obvious patch where the bare wood is showing. I have no idea if it has not taken to the oil or if the seller has sanded over it too much to reveal the wood. They said it was solid teak. I'd need to fix this patch too. I will attach a pic. 
 
I'm going to go with your advice and buy some #0000 steel wool and non-varnish teak oil. Is this the right one? I don't know if its varnish free or if they ship to UK...
 
http://www.amazon.com/Star-Brite-Teak-Oil-Quart/dp/B001AJ7O5O
 
Do I have to go over the entire table with steel wool and teak oil or can I just do the areas that need doing? Or will it will bad and uneven? 
 
Thanks for the tip about soap for cleaning. And if I use the beeswax on top does that make the surface permanent and harder to reapply teak oil again? I read some use mineral spirit to remove stains and then rub teak oil in small areas with a lint free cloth. Would be interested in your views on this? Mind you the iron trick they advised has caused me a near burn mark on the table!
 
I'm going to be very careful from now on. Just a damp cloth and soap if needed. Hopefully when its done I've thought about covering it with a table cloth but its a shame to cover up a nice piece of furniture. 

The first pic below is the area thats sticky and a small patch where the bare wood is showing.
The second pic shows the lighter marks after I used a sponge and detergent over it. 

Lastly the cup water mark and the near burn mark

If you're going to use the table as an ironing board, it may be wise to invest in a table pad.  
Table pads suck, but they sure do protect the table.

Ha ha. I used table pads actually. That was listening to advice on numerous forums to use an iron and linen cloth to get rid of water marks. And the water mark was made from a hot cup resting on a table pad/placemat. 

The photo with the lighter patch shows substrate appearing where the face veneer has been sanded or scraped through.  I'm sorry to report that there is no repair for this defect, other that a skillfully painted disguise.
On a separate issue, it should be clear that oil would not penetrate a waxed surface -- for what that's worth.

Star brite doesn't do the trick, since their's is a product for patio furniture. Indoors furniture need indoors furniture treatment, and a vintage danish table is better suited with a finer product (that is milder, both to the wood and the environment) Past advises still apply!

Chronophage, have you used Star-brite teak oil?  Is it available in Norway?  It's just linseed oil with a minimum of solvents in it.  Yes, it's marketed for boats and outdoor furniture but it's certainly fine for use on furniture, too.  I've used it for years with beautiful results.  I am sure there are similar products in the UK, though--no need to pay high shipping costs just to get Star-brite.
To the OP--can you get some oxalic acid?  A lot of us here have had excellent results using a very dilute solution to remove dark marks on teak (and walnut).  It might work on the area that you nearly burned with the iron.  I had a teak table with a large dark area with blurred perimeter (not the kind of black stain from a spilled liquid or wet glass or flower pot).  It was there when I got the table so I had no idea what caused it.  I made a slurry of water and Barkeepers Friend (mildly abrasive cleanser with 10% oxalic acid) and rubbed it on the stain for maybe a minute, then rinsed off thoroughly and dried immediately.  The stain was completely gone---looked as if it had never been stained at all.  
Other people here have left the oxalic acid on for much longer periods but I have gotten good results with just a minute or two on some stains.  I've also gotten great results leaving it on longer, but this sometimes ends up bleaching the wood too much.  So now I start with just a minute or two and see how that goes.
The big lighter area may just need oil.  I agree with SDR about the small light patch--looks like the veneer is sanded through.
The sticky area may be wax that is gummed up from something else, like oil or detergent...?  Or wax that never properly dried becuase it was applied too thickly?  Hard to say.  I have read that turpentine is the only solvent that will dissolve wax so that's what i've always used to remove it.  

No, spanky, I have not (yet had the pleasure:) -; touché indeed …

Thanks again for the advice everyone. I'm really confused what to use now.  There seems to be a lot of conflicting information about teak oil vs Danish oil. I don't want anything pigmented or toxic as I have a young child although they say they are toy safe. I'm considering Liberon Superior Danish oil (inc UV filters) as recommened in the past thread, or Ronseal Colron refined Danish oil ???? but still non the wiser.
 
So let me get this right. 
1. Use oxalic acid to remove the burn mark area. Not the whole table top? 
2. Leave teak/danish oil on the whole table top for 15 mins. Then use steel wool. Wipe off with dry cloth. 
3. Another layer of teak/danish oil and wipe off with dry cloth. 
4. Use wax over the whole table top.  
5. Maintain with wax once a year. Or to re-oil take off wax with mineral spirit and then re-wax.
 
So I think the seller has tried to disguise their mistake of sanding through the veneer by using some type of sticky varnish to mask it - and it obviously hasn't covered it up. Its a small area about an inch wide but as it is near the centre of the table it is noticeable and its annoying me now I know what it is. 
 
Does anyone have any tips on masking it? Crayons? Oil paint? 
 

As I wrote earlier, paint is the only way to cover the revealed cross-grain substrate.  And no one in the world can make one thing look exactly like another using foreign substances, so the repair will only be an improvement.  It might be easier to keep the damage covered with a centerpiece or a place mat -- and move on to other problems.
Paint cannot be applied over oil or wax, so the painting should be done first.