How do you clean the smell of smoke out of a cedar chest?

Remove all contents from the cedar chest.

Sand the interior wood with fine-grit sandpaper. ...

Wipe the inside to remove all dust left from the sanding.

Place an open box of baking soda inside the chest and close the lid for 48 hours.

Then vacuum out the baking soda. You could also add an essential oil to the baking soda such as lavender, clove or lemon.

What Dee said. Paint would work as well on the outside but a sanding and refinishing of the whole chest would be the quickest way to eliminate the smell.

I wouldn't put a box of baking soda in the chest. I would liberally sprinkle the baking soda in the chest covering every knook and cranny. For the outside you can seal the odor into the wood.

How to Eliminate Odors in Wood

Your best hope for sealing in any odors is shellac. It makes an excellent barrier coat over all wood species, and it sticks to almost anything. Zinsser makes a great product for this purpose, called Bullseye SealCoat. It’s a clear dewaxed shellac that you can apply to those drawer interiors easily with a brush. It dries quickly and should help to diminish mice, mildew, musty, smoke, urine smell. And since it doesn’t contain wax, you can apply other finishes like lacquer or varnish over it — they’ll stick just fine. SealCoat is also handy to have around the shop as a general purpose wood finish and sanding sealer. So, I’m sure you’ll find other uses for what’s left of the quart you buy.

I've found that putting things out in the sun is a great way to get rid of smells, though I will admit I haven't tried it with wood specifically. If the other suggestions don't work, try putting it outside on a sunny day. Make sure it's open so the sun can get to the part that smells.

OdoBan is an excellent tool to remove smoke odor. But first I would try lightly sanding the interior of the chest. This will release the original cedar smell. Then wipe down the exterior.

Wash it out with a sweet smelling disinfectant and then dry it in the fresh air!