Step 1: Get Some Metal Salts
Different metals will burn different colors. Your homework assignment is to provide a detailed explanation why this occurs, based on first principals of quantum chemistry. For now please accept the following conversion chart:
blue = copper chloride (CuCl2)
green = copper sulfate (CuSO4)
red = strontium chloride (SrCl2)
purple = potassium sulfate (K2SO4)
white = magnesium sulfate (MgSO4)
In my experience copper chloride works the best by far, giving bright blue flaming pinecones that are sure to impress even the most discerning pyromaniac. The green and red from CuSO4 and SrCl2 are also attractive. Magnesium sulfate (white) and potassium sulfate (purple) didn't do much of anything to the color of the fire for me, so I'd say skip those two. My sources tell me there are other combinations you can try.
Of all the chemicals listed above, only MgSO4 can be purchased at the drugstore (in the form of epsom salts; magnesium sulfate heptahydrate). For the rest you'll have to do a little hunting. Strontium salts are commonly found in road flares, for example. Suppliers for pyrotechnics, photography and rocketry are also a good source. Google product search turns up a surprising number of results for copper chloride. You may find these salts in their anhydrous or hydrated forms, which doesn't matter since you're going to dissolve them in water anyway.
Note that all of these things are bad for you if ingested, so be careful to wash your hands after handling, and try not to spill the CuSO4 in your coffee. Gloves might be smart.
Step 2: Prepare Coloring Solutions
Add the pinecones and let them soak for at least 2-3 hours, preferably overnight.
Step 3: Dry the Pinecones
Once the pinecones are good and dry you are ready for action.
Step 4: Set Them on Fire
A final safety note: It's always a good idea to have a fire extinguisher on hand when you're burning things. And don't breath the fumes from the colored flames.