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.
So what does everything burn to?
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.
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?
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.
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)