Do chemical bonds have mass? When an exothermic reaction occurs, the energy in the chemical bonds of the reactants is partially transferred to the chemical bonds of the products. The remaining energy is released as heat.
For example:
$$\mathrm{N_2 + 3H_2 \to 2NH_3} \qquad \Delta G^\circ = -32.96 \,\rm kJ/mol$$
Therefore, when $1\,\rm mol$ of nitrogen reacts with $3\,\rm mol$ of hydrogen (under standard conditions), we get $32.96\,\rm kJ$ of heat.
Now, applying $E=mc^2$, this works out to be
$$m = 32.96 \times (3 \times 10^{-8})^2 = 2.96 \times 10^{-14} \,\rm kg \quad \text{or} \quad 29.6\, pg$$
Does this relationship hold? Do the products of an exothermic reaction really weigh ever so slightly less than the reactants?
In a more general sense, does removing energy from a system decrease its mass (or vice versa)?
 A: Yes, bonds have mass, like every other kind of energy.
This can be significant; if you had a glueball (a hypothetical particle made of massless gluons), it would have mass, and all of the mass would be from the bond energy! Same would go if you somehow managed to bind photons together. 
A: As far as the theory goes, you are absolutely correct, the (negative) binding energy between atoms in a molecule contributes to the total mass of that molecule, so a stable molecule is less massive than the sum of the masses of its constituent atoms.
However (as you yourself calculated), the mass difference is absolutely tiny, and as far as I know, it has never been measured. But the principle is no different from the mass deficit that occurs in nuclear reactions and that, in turn, is readily measurable. Consider the atomic mass of deuterium ($2.01410178\,\rm u$) vs. helium ($4.002602\,\rm u$), which is about $0.64\%$ less than the mass of two deuterium atoms. The difference is the energy that would be released in a fusion reaction.
So yes, in general, removing energy from a system decreases its mass, and conversely, adding energy to the system increases its mass. The most extreme example perhaps would be protons and neutrons: roughly $99\%$ of their masses come from the (positive) binding energy between their constituent quarks, and only about $1\%$ is attributed to the quark rest masses.
A: I realize this isn't the main point of your question, but it seems worth mentioning that the Gibbs free energy change $\Delta G^\circ$ is not a measure of the amount of energy released in the reaction. It's an abstract quantity related to the entropy change of the universe. 
The relevant amount of energy released that you would use in the mass change calculation is the change in "internal energy", which for a reaction at constant volume and pressure is equivalent to the enthalpy change $\Delta H$. 
 In your case where the number of moles of gas is decreasing, we would need more information about whether pressure or volume or neither is held constant. 
A: I know I am too late for this question but I couldn't stop myself from answering.
I don't think that chemical bonds store energy. So saying energy in the chemical bonds can be misleading.
Bonds are not batteries. The energy is rather stored in the atoms.
Look when two atoms combine , they release energy and that energy can be of any form . It could be even absorbed by surrounding atoms or released as photons. Those energies are not stored in bonds. Bonds don't have any physical appearance. So to break (or separate) a chemical in it's constituent atoms , you need to give energy and that energy is then absorbed by those atoms and they (the atoms which formed bond earlier) just move apart .
Also for exothermic reaction to occur , first of all  you need to break the chemical (and the energy needed is called activation energy) and those atoms then recombine and again they release energy and the amount of this released energy depends upon the pattern of rearrangement of those free atoms.
That's why energy of reaction is defined to be the change in energy of formation of products and reactants.
Depending upon the amount of energy the reactants need to go apart and the energy the separated atoms release by forming a new compound, we classify the reactions as exothermic and endothermic.
And now coming to your question :
Since energy is released by the atoms, so they have now a lesser mass than what they had when they were separate. And that can be calculated using the well known equation :
$$ E = mc^2$$
Hope it helps .
