Advice: Refinish Arne Vodder Desk  

I've refinished quite a few...
I've refinished quite a few of these desks over the years.
First establish if those marks arent simply on the surface.
Take some super fine steel wool and some furniture oil and see if the marks rub out.
If they dont then youre going to need to refinish the desktop.
These desks usually have a very light varnish so they only require light sanding.
Start with some 120 to 150 grit sandpaper and carefully sand until you remove the varnish. Then you can sand a bit more on the areas that have stains. After you get the desired effect work your way up to some 220 grit sandpaper.
Apply a few coats of Natural Watco Danish Oil (very close to original finish) and after drying polish with some super fine steel wool and paste wax.

Thank You!
That is exactly the kind of advice I was looking for - I just didn't know where to start.
I appreciate it!

It looks like the rest of the surface is in very nice condition, lots of nice dark striations to the timber without much wear or fading.
You need to establish what sort of finish is presently on the desk, this could be anything from oiled or wax finishes to some form of lacquered polish.
If it is the former you could try lightly cleaning the surface with fine wire wool dipped in White spirit or Meths and then re-applying a good quality Teak oil. This is often all that is needed with light marks such as this.

I would lean more towards the...
I would lean more towards the steel wool reconditioning recommendations than sandpaper refinishing one. I would only remove the original finish on a nice Vodder piece as a last resort. And nothing I see from those photos would indicate that it is in last resort condition.
My favorite product to use is Howard's Restore-a-Finish, available in most hardware stores. While it has a hokey-sounding name, it can do a remarkable job with 0000 steel wool in eliminating scratches, discoloration, and surface marks. Even if the marks are not completely eliminated, they can often be removed to an extent that they are no longer a visual detriment.

Put a book or lamp on the...
Put a book or lamp on the desktop and you're done, seriously, the marks are on the rear of the desk and it's against a wall. Or you can buy some furniture touch up markers at the grocery store on the corner.
If you do actually properly refinish the top, have a piece of glass cut to fit it if you're going to use it with any vigor.
Let's hear it for plastic laminate!

Great!
Thank you everyone, tons of good advice!