One of the reasons for using teak -- especially for outdoor furniture -- is its abundance of natural oils. The oils prevent rot and give the wood a shiny patina, but they also inhibit paint adhesion and can bleed through finishes, so painting teak isn't the best idea in the world. You may have some success, though, if the wood is old and weathered. Proper preparation is essential, and it doesn't hurt to seal the wood before painting.
Wash the Wood
You don't have to worry about sanding off the layer of gray wood that discolors old teak that has been outdoors, but you do need to wash the wood thoroughly to remove dirt, excess oil and mold that can interfere with paint adhesion. This is a job for a strong detergent; mix 1/2 to 1 cup of trisodium phosphate in a gallon of warm water and wipe down the wood with it, using an abrasive sponge. Wear rubber gloves and goggles when working with TSP, and rinse the wood thoroughly with clean water after you've washed it.
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Fill Holes and Scuff
Sanding teak before painting it isn't necessary, but as additional insurance against peeling, you should scuff the wood with 220-grit sandpaper. Before doing this, fill gouges and holes with epoxy wood filler, so they won't be visible after you paint. To use epoxy filler, mix it with the hardener that comes with it in the recommended proportions and spread it with a putty knife. Use it within 5 minutes of mixing to ensure it remains pliable, and sand it flat with 120-grit sandpaper after it sets.
Apply Stain-Blocking Primer
The natural oils in teak -- like the sap in knotty pine -- will bleed through a paint coating and need to be sealed with a primer. Some high-solids latex primers will do the job, but to be on the safe side, prime with a shellac- or lacquer-based product. Apply the primer with a natural-bristle paintbrush, brushing with the grain of the wood, and let it dry before touching the wood. If you can, wait for a week before proceeding to see if the oils bleed through the primer. If they do, apply another coat before painting.
Paint at Least Two Coats
Even if you tint the primer a color close to that of the paint you're using, you'll need at least two coats of paint. Choose either latex or oil-based paint -- both can be applied over shellac-based primer. Brush a single coat with the grain of the wood, using a synthetic-bristle brush for latex paint and a natural-bristle brush for oil-based paint. Scuff the first coat of paint after it dries; then apply a second coat in the same way. To give your paint job extra protection, finish off with an optional coat of clear polyurethane.