How can I darken the color of painted furniture using varnish?

Varnish has a yellow cast to it and yellows over time. It would not darken the color. it's just a protective finish.

Maybe dry brush a darker gray over it?

I would recommend using antiquing wax. That will darken and add dimension to your finish

You could use a dark antique wax or paint it a darker color before varnishing

If its a painted set then a dry brush of a slightly darker color would definitely work. If its stained you could consider going over it with a gel stain - make sure and wipe it down thoroughly and then apply your varnish.

If you've only just painted it, you can paint it darker. There are also ways to use an antiquing faux finish or glaze effect. Varnish is not going to darken, it's a sealer coat. Once something has been sealed, it's more complicated to darken it.