My varnished doors are old & have become gummy. How can I clean them?

http://www.howtocleanstuff.net/how-to-clean-varnished-wood/

Use a paste of baking soda and a magic sponge. Works wonders!

soft nail brush - Sponge and soapy water... Plus old cloth to wipe it down with.

I’ve used Watco natural oil and a fine grit steel wool. Pour oil on the steel wool to keep it saturated and rub the steel wool with the grain of the wood. Long as you keep a good amount of oil scratches won’t show and it will lightly sand all the foreign materials off your door and just wipe excess oil and the gummy substance off with clean rags! It’s important not to wrinkle up used rags saturated in oil and throw in the garbage. You must lay out all used rags flat on a area where there’s no combustible materials near your rags! Any used rags saturated in oil if piled up or wrinkled up in a ball can build up enough heat to spontaneously combust and start fires. This is a very real threat you need to take precautions for your own safety and protection of your property. I use to think this was a joke that these rags could catch fire till I seen a rag burst into flames before my very eyes! This is no joke! Please be careful when using any kind of finish or stain with rags, any of them can spontaneously combust if wrinkled or piled up! Always lay out flat and let dry completely before throwing them away I. The trash! Once their dry their safe to throw away

My first try would be wiping with a cloth soaked with mineral spirits type of paint thinner.

i can attest to the fire issue. Rags caught fire in my basement. No, rags are kept in water.

https://www.consumerreports.org/cro/magazine/2013/01/three-cleaners-remove-haze-from-dishware/index.htm