Does fire have an influence on the weight of a burning object? I know that fire is a chemical reaction, and for that reason it has no mass, but I'm not sure that this fact is enough to conclude that it has no effect on the weight of a burning object. Consider the case of a balloon, obviously in this case, fire has an effect because of the well know fact that a balloon achieves buoyancy by heating the air inside, but I really want to know if it has an effect on a burning building for instance, does it weigh more or less?
 A: The answer depends on the definition of the system.
Imagine a piece of paper burning in air to produce carbon dioxide and water.


*

*If the system includes just the paper the system will lose a lot of mass as CO2 and water are given off.

*If the system is the paper, CO2 and water in a glass bowl in thermal contact with the rest of the universe, the system will lose a tiny amount of mass as a result of energy from combustion being transferred to the rest of the universe.

*If the system is 2. but enclosed in a sphere which is perfectly reflecting at all wavelengths and thermally isolated from rest of universe, there will be no mass change.
A: Chemical bonds, just like nuclear bonds, have an associated mass to them.
Nucleons make up most of the mass of the atom, with about 938 MeV per nucleon. Nuclear binding energy is usually on the scale of a few MeV, meaning that the mass difference during a nuclear reaction will be usually under 1% of the mass of one proton.
Chemical reactions, on the other hand, are usually on the scale of a few eV. Meaning that a chemical reaction's change in mass will be about one million times weaker than a nuclear one. That's why you can pretty much always ignore any effect having to do with the chemical bond mass of an atom.
A: There are three things to consider here:
Mass of combustion products
Since combustion is by definition the chemical combination of the fuel with oxygen, the mass will go up. If we burn a material with a volatile oxide, such as carbon, the mass of solid material remaining will go down (but really the lost mass is present in the atmosphere.) If we burn a material with a nonvolatile oxide, such as magnesium, the mass of solid material will go up. 
Relativistic decrease in mass due to release of chemical energy
This is covered in Slereah's answer. It is generally negligible.
Updraft
Combustion releases a lot of heat, which reduces the density of the combustion products and surrounding air causing buoyancy and hence convection currents. This is the reason a candle flame points upwards.
Updrafts are strong enough to carry small particles. You will see small glowing red particles carried away by the updraft of a bonfire, and rather more by the updraft of a wood fire in a hearth with a chimeney. Smoke is just small particles of material (typically unburnt charcoal or burnt ash) carried away from a fire by the air currents it produces.
Thin tissue paper when burning, can create sufficient updraft to lift the whole sheet. This can be quite scary if you do it accidentally indoors.
So fire doesn't change the weight of anything, but it does create air currents that are capable of lifting objects.
