Is the molar mass of a molecules lowered by the binding energy of the chemical bonds?
The difference is the best observed for products of nuclear reactions, related to nucleon binding energy due the strong force.

By comparing the mass of $\pu{2 mol}$ of $\ce{^1H2}$ and $\pu{1 mol}$ of  $\ce{^4He}$, the difference is about $0.7\%$. 

The chemical bonds have difference more then million times smaller.

E.g. if there is an exothermic reaction producing $\pu{500 kJ/mol}$, the mass difference per mol is $$\Delta m = \Delta E/c^2 = \pu{-5.56 ng }$$

That means, we have hardly ever scales with the needed accuracy.