Can I wash and continue to use dishes that were in a sink with a brush used for varnishing?
Varnish is usually a mixture of a drying oil, some resin such as pine tar, and a solvent such as turpentine.
The drying oil most likely won't kill you. The most commonly used oil, linseed oil, is actually just non-food-grade flax oil by another name.

The Greeks put pine tar in their wine, and it's a tasty treat.

The solvent is a little more troublesome. The pinene in turpentine for example, has a nasty looking chemical structure. However, Solvents, by their liquid nature are pretty easily washed off, epecially when you put plenty of soap in your water.

I'd give the dishes a good washing, and use them without a second thought.
I'd also try not to leave a paintbrush in the kitchen sink again.