I started small. I began by spraying down both dressers entirely with vinegar,
and left them outside overnight to air out. Usually this step is enough to get rid of most musty smells, but it did nothing for the cigarette smell.
The next step was to try wiping them down with a TSP substitute. Wipe it on, let it sit for a few minutes, and then wipe it off.
Again, nothing. Darn.
So I pulled out my trusty shellac. The zinnser shellac and (this primer) are widely used odor blockers. They are also very hazardous, so I use them as a last resort.
I was planning to distress these pieces after I painted them, so I didn't want to use a pigmented odor blocker.
The clear shellac can be applied with a brush, roller or a rag (or sprayed). I opted for a lint free cloth. It dries very quickly so you have to work fast to avoid any drag.
The minimum recommendation to block odors is two coats. The smell was so strong for these pieces that I applied three to every inch (inside and out) before the smell was gone.
Lightly sand with a 220 grit sandpaper before applying paint and you're good to go!
(Clear shellac is also a tannin bleed blocker. If you're experiencing any bleed as you're painting and don't want to prime with a white primer that will show through when distressing - this shellac works great).