What happens to the mass of a burned object? If I were to burn a pile of wood weighing a hundred kilograms and I would have a big sack hanging over the burning pile. In this sack I would catch all the smoke that came from the burning pile, if all the wood turned to ashes and I'd put this in the sack with smoke. Would the sack weigh a hundred kilograms or would it weigh less?
Is it the case that all the mass of the burning pile is converted in to ash and smoke and therefore weigh the same as the unburnt pile of wood? Or is it the case that because of $E=mc^2$ the burning pile emits energy and the mass of the pile is converted into the heat that a burning pile of wood gives and therefore takes away most of the mass?
 A: Other answers have focused on the combustion products mass, and the mass of oxygen. I want to cover the other part not in those answers.  I'm also keeping it very simple, so.bear in mind this isn't how a Chemistry or Physics graduate would describe it. I'm ignoring truly tiny effects that only get noticed at degree level and higher, and being a bit simplistic.  I think from the question, simpler is what is wanted. Please bear that in mind when you read.....
E = mc2 is an equation showing matter-energy equivalence.  In the situation you describe, you're trying to use it as an equation concerning matter converted to energy, and therefore presumably how some mass could be "lost" from your "bag" of combustion products.
In general, and in simple terms, chemical processes such as reactions, and combustion (burning), and physical changes of state (boiling/evaporation), do not involve matter-energy conversion. What is happening in simple terms is, that chemical bonds are being reconfigured due to the presence of other chemicals, or physical conditions. The chemical bonds usually involve the outer electron/s of atoms, and their bonds to other atoms.
If a different layout (configuration/bond) is energetically favoured, and achievable, then the existing configuration will change to it. The resulting bonds may need less energy, in which case the reaction gives off heat (such as combustion processes, or more generally exothermic processes - ones that give off heat).
But that heat isn't obtained by converting matter to energy. It was energy already, stored in the form of higher energy chemical bonds (a type of "potential energy"), that became changed to lower energy chemical bonds, and the "spare" energy was released as heat (a type of "kinetic energy").
If it helps, also imagine a tall stack of bricks. When it topples, the books were stationary but now move quickly (kinetic energy). But that speed didn't come out of nowhere, or by converting matter to energy.  It was originally energy at the start - the bricks were higher up in the Earths gravitational field, and this was also a form of potential energy. What has happened to the bricks is, the potential energy of their higher position is now a lower.energy position so they have lost potential energy, and the potential energy they lost, was converted to kinetic energy - speed. Likewise with your combustion - the burning processes involve chemical bonds moving from higher states to lower states of potential energy. The potential energy they lost, was converted to kinetic energy in the form of heat, this time.
(We can follow this in the books example a bit further - when the books land on the ground, their kinetic energy is dissipated as friction and vibration (of the ground and air), which both ultimately end up as very low grade heat. Ultimately the room becomes an undetectably tiiiiiiiiny bit warmer! But still its energy changing form, not matter changing to energy)
A: You would have much more mass than 100 kg after the wood was burned.  As it turns out, wood is made of cellulose and lignin.  Both are cross-linked glucose polymers, so a good approximation of what you would get is given by the chemical reaction of burning glucose:
$$\rm C_6H_{12}O_6 + 6O_2 \to 6CO_2 + 6 H_2O$$
This means that 6 oxygen molecules combine with one glucose molecule when it is burned.  The molar mass of the glucose molecule is 180 and the molar mass of the six oxygen molecules is 192.  This means that when you burn 180 kg of glucose, 192 kg of oxygen take part in the chemical reaction, producing an equal mass of carbon dioxide and water vapor.  At these ratios, when you burn the 100 kg of wood, you would collect 207 kg of carbon dioxide and water vapor.
A: Greetings from Mikhail Lomonosov from 18th century.
In chemistry (and burning is pretty much a chemistry) mass is considered a conserved property.
(It is a conserved property outside of the chemistry as well, but chemists have the luxury not to deal with the heat energy used or produced, because it has a negligible mass in chemical reactions.)
If you have a closed (and heat resistant) container with the wood and all the air (or oxygen) needed to burn it down, the mass of the container will not change by burning the wood.
What most people skip in these considerations is that gases have mass. A cubic meter of air in normal conditions is like 1.2kg - not much, compared to a 1 cubic meter of piled wood (like 600kg), but still important.
A: Depends on what your sack manages to capture
This was a thing that finally helped kill the phlogiston theory of fire (that burning something means releasing the phlogiston enclosed in it): most things got lighter by burning them, but some got heavier! This could only be explained by having some materials contain negative-mass phlogiston, which pretty much everyone agreed was silly.
Reaction products
So what does everything burn to?
Metals
When you burn a metal, you get a metal oxide. Most metal oxides are not volatile, and they're the white ash you're left with when you fully burn wood. Metal oxides are heavier than the metals or metal ions you started with, because you've added oxygen.
These ashes can actually be a good source of metal oxides for making things such as lye.
Carbon
When you (fully) burn hydrocarbons, you get carbon oxides: Carbon dioxide and/or carbon monoxide. Both of these are gases, though you can capture them with an airtight bag. If your fire isn't enclosed, your wood smoke will also contain various volatile hydrocarbons (due to pyrolysis) that still have various other elements attached to the carbon.
If your fire doesn't quite fully burn, you'll also be left with "black ash", which is actually just charcoal, or mostly pure carbon.
Again, if everything is fully burned, the resulting products will be heavier than the starting materials, but can you capture them?
Volatiles
Wood also contains hydrogen, oxygen, nitrogen and (in smaller quantities) various other (non-metal) elements. When burned this turns into water (vapor), free nitrogen and various other gases. The water is fairly easy to capture. The other gases will be harder. All of these are heavier or the same weight as their starting material.
Results
If you can capture everything, you can certainly measure that the contents of the bag are now heavier than the wood you burned. This is only logical, as the bag now contains the wood plus all the oxygen from the air that you used to burn the wood.
If you cannot capture all the gases, the answer will depend on what it is you burnt. For most materials (such as wood), the contents of your bag will be lighter than the starting material. For metals it will be the opposite. The answer for any material in particular will depend on the ratio of capturable and non-capturable materials, along with how much oxygen the capturable materials will bind to them by weight.
Fun fact: for wood, this is called the "ash content". For wood, this is typically 0.1% to 0.2%.
As $E=mc^2$, the energy involved in chemical reactions is far too small to be measured outside of a laboratory setting. You will lose a tiny bit of mass-energy due to energy being released, but it will be less than a speck of dust that you didn't quite manage to get onto the scale, or a fingerprint you left on the bag. (I mean that last bit literally. A fingerprint is about 50 μg, which has a mass-energy of about 4.5 GJ. That's about a quarter ton of dry wood worth of energy)
A: From The Evolution of Physics:

... all energy resists change of motion; all energy
behaves like matter; a piece of iron weighs more when
red-hot than when cool ...

In answer to your main question, energy from the chemical bonds is converted to energy in the form of heat, so there won't be a change in the mass of the products of combustion and the result unless you allow the gases to cool.
The process is more complicated though but to create a simple example let's take David White's answer.  100kg of wood at room temperature will combine with 107kg of oxygen at room temperature and create 207kg of hot gases.  In an isolated system, there is no net change of mass since the energy of the chemical bonds is converted to heat energy.
If you allow those gases to cool back to room temperature, they will have less mass because that heat energy resists change of motion like matter.  Looking at wikipedia, red oak has an energy content of about 14.9 megajoules per kilogram.  100kg would then be 1.49E9 joules.  Also according to wikipedia the conversion rate is about 8.99E16 joules per kilogram.  Dividing that out I get 1.657E-8 kilograms or 1.647E-5 grams in the form of heat.
So cooling that 207 kg of hot gases back to room temperature would make it lose about the same mass as 1/2000th a grain of rice.  That energy (mass) isn't 'lost' however, for example some will be emitted in the form of black-body radiation
A: Relativistic loss of mass is unmeasurable here, but in principle, you’d lose some tiny fraction of the mass by heat transfer to the surroundings.
Whether the smoke would weigh more or less than the wood depends on your definition. Oxygen from the air is combining with carbon in the wood to form carbon dioxide. If this counts as smoke, then the smoke weighs more than the wood because it includes the weight of the oxygen. If smoke is just the particulate stuff you can see, then it weighs much less.
