The chemical energy makes up a small portion of the mass of the stick of dynamite. $E=mc^2$ does not apply for just nuclear reactions! Nuclear reactions are the archetypical way of "verifying" the equation though in a popular physics sense. (Only about <0.7% of the mass can be turned into energy in a nuclear process. I know 0.7% is the value for fusion, fission should be less.) (Actually, Einstein proposed massing radium salts before and after some nuclear decay in 1905, long before fission.) There are lots of other ways to see $E=mc^2$ in action. Here are a few (including your example):
1) Chemical bonds. What we usually think of chemical energy is contained in the chemical bonds between molecules, etc. This is energy, and by $E=mc^2$ can be realized as a part of the mass of the chemical system. So when the bonds are broken and formed in a dynamite explosion, some of that mass is lost as radiant energy (infra-red radiation, i.e. heat).
2) Gluons. This is really interesting, at least to me. I'm sure you've heard of the Higgs boson and the Higgs mechanism. In popular literature it is said to give all things mass. But the interesting thing is that if you look at the mass of a proton (~938MeV) and masses of the constituent 2 up-quarks and the one down-quark (each ~4MeV), we see a HUGE disparity! Where is the extra mass? Here $E=mc^2$ and quantum mechanics come together to tell us that in the proton there is actually a sea of virtual quark-antiquark pairs and gluons. (This is called the dynamical model of the nucleon). The energy contained in all the gluons and virtual quarks makes up 99% of the mass of the proton.
3) Gravitational radiation. This requires some understanding of general relativity. In a binary system, the quadrupole moment is nonzero. Therefore, the system emits gravitational waves. Those waves have energy, so the mass of the system should decrease. This corresponds to an increase in the rotation frequency of the system. This effect has been measured to a very high degree of accuracy (no numbers off the top of my head).
4) Accretion tori. We can use accretion tori around black holes to convert mass into energy by exploiting the gravitational binding energy. This effect gives about 6% for a nonrotating black hole and 42% for a rotating one.