How to get nicotine and coffee stains off dentures?

See your dentist. Over the counter denture cleners and pastes will help, but your dentist and dental have ways to do a better job.
Try your dentist! Coffee and nicotine stains can be scrubbed off but you have to be careful not to abrade the denture's smooth surface. We use an acid rinse in an ultrasonic cleaner that works wonders. I recommend you ask your dentist to clean your teeth the next time you go into have them checked, which by the way, is something all denture wearers should do.
Good Cleaning. If commercial denture cleaners do not remove consider scrubbing dentures with toothbrush with baking soda and salt mixed with water forming a paste.
Use baking soda. Brush with baking soda and a stiff brush soak overnight in a cleaning solution like efferdent.
Tooth brush and past. Use a paste of tooth paste and asking soda to remove any stubborn stain that denture loaners do not remove.
Clorox. But be careful to use it only on the teeth, not the pink base.
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.