How do I refinish an espresso bed frame to match an antique dresser?

A photo would help. Then we can make suggestions as to what products might work. And furniture doesn't need to match either.

It is mainly the shape of the piece of furniture, rather than its finish that bespeaks its style.

Take a Photo of the Antique. Take this photo with you when you are selecting Paint Colors. If you want an exact match you will need a custom mix because all antiques yellow or cream color in time, (hence why Antique White differs from White White). Likely you find something that is a close enough match to secure you don't need to paint both.

I would use a gel stain! Check out General Finishes Gel stains they should have the right shade of gel stain as your dresser. :)

https://generalfinishes.com/wood-finishes-retail/oil-based-wood-stains-sealers/oil-based-gel-stains

You could chalk paint it and then distress it! Here are some tips: https://www.aratariathome.com/blog/chalk-paint-tips