What's the best way to clean coffee and tea stained dentures?

Not toothpaste. Soak in denture cleanser and use soap along with a soft brush. Toothpaste is too abrasive and will be hard on the acrylic surfaces in most dentures. If the staining is persistent or severe, ask the dentist to use an ultrasonic cleaner on them when you are in for your annual oral cancer examination.
Dentist. Go in for your checkup and ask them to clean your dentures. Sometimes they will be ultrasonically cleaned and then polished. Depends on office. We have much stronger cleaners for this. At home, clean thoroughly with soft toothbrush and toothpaste, then soak in antimicrobial denture cleaner.Some of my patients really like "stain away" as well as the effervescent cleaners.
Ultrasonic cleaner. Have your dentist run it through the ultrasonic cleaner which most offices will have. Don't try to clean it with toothpaste as the abrasives will affect the acrylic and then spoil the fit of the denture.
Manually brush. Soaking is fine, but to attempt to really clean the dentures, the dentures need to be soaked and then scrubbed (but not with anything too abrasive to leave scratches, etc on the dentures. Sometimes the stains are so extreme, you may want to bring to a dentist and he/she may have to send to a dental lab.