We have bought a house that has nice wooden doors, but holes in them have been filled ready for painting. We would prefer to varnish the wood - what can we do about the white filling in the holes?
Test it first. You may be able to just stain the filler as is.
If that does not work, dig out the existing filler with whatever tool works (putty knife, old drill bit, a thick nail) and replace it with something you can stain. Not sure if paint stripper works on filler, but it's work a short.
If there's any other stains or treatments on the door though, you need to remove those with chemicals like The Evil Greebo, and sand it down nice and fine.
Assuming they used a proper, stainable wood filler, your best result will occur if you strip the wood to remove any existing finish, and then re-stain the wood including the filler, so that you get a consistent look to the wood.
The downside is, stripping wood involves the use of pretty harsh chemicals, and a lot of manual labor scraping and then scrubbing with steel wool. But the end result will not be better using any less intensive method.
Consider faux finishing the spots. You can find many good sources for simple approaches to creating a painted finish for those spots that would mimic the wood. This is an answer if the filler will not take stain. Paint the spots carefully with a good primer. Using small brushes and different shades of glaze (a semi-transparent paint) lay down layers to complete the grain that is missing. If you look at the grain you can see it as layers of different shades. Probably two or three would do it. Light, medium, dark, allowing each to dry before applying the next. This is not as difficult as one might think. The thing about decorative painting is that no one will ever look closely or even expect it to be perfect. The goal is to get the eye to go over it without stopping. It doesn't need to be great. Some judicious smudging can go a long way to help it look natural. Have fun.