When atoms get heated up, they start emitting EM radiation because of accelerating electrons. But why don't the protons emit EM waves which destructively interfere with other emitted EM waves, because they're also accelerating just as the electrons do, because they're both jiggling as one (atom)?
The existing answer is correct, but it's in a sense 'orthogonal' to the question, and it doesn't really address the core concern.
Atoms are indeed quantum mechanical objects, and the classical understanding in terms of accelerating charges no longer works, and it needs to be replaced with transitions between different states of the atom, but this does not make the problem go away.
The reason for that is that the transition has an amplitude - a number which governs how likely the atom is to change states, known technically as an electric dope transition matrix element - which depends on the position of the electron with respect to the center of mass, but also in the position of the nucleus, which therefore also contributes to the radiation, exactly as in the classical case.
So, why doesn't the radiation cancel out? Well the word dipole should give it away: what really matters for the radiation isn't so much the movement of charge, but the dynamics of the electric dipole, and here the contribution of the nucleus adds constructively with that of the electron: it has the opposite charge, sure, but it also moves in the opposite direction, so the global sign is $+$, and the two contributions to the radiation / acceleration / transition dipole / whatever are in the same direction.
Moreover, this is also the case for the classical dynamics of the Rutherford model of the atom: the nuclear charge is opposite to that of the electrons, but it moves in the opposite direction, and the contributions add constructively. This is why I said the answer "because QM" is orthogonal to the question - the answer doesn't really care whether you're doing classical or quantum dynamics.
Now, why do we normally ignore the nuclear motion? Well, it is much heavier than the electrons, by a factor of ~1800 and up, so it doesn't move much, but it still has a contribution and its effects can be seen in the atomic radiation if you do a careful enough experiment. However, because its effect is constructive, the answer wouldn't change even if you had two opposite charges of the same mass orbiting each other (as in, say, positronium).
The classical idea that the electrons emit energy because they are accelerating does not work in the atomic realm. This was the precise reason that Rutherford's atomic model was rejected by the classical physicists in the early 20th century.
The atoms emit light when the electrons from one energy level make a transit to a lower energy level due to the instability of the atom. The same happens with the case of the proton too, but the energy gap between the consecutive energy levels of the proton is so large that the thermal energy cannot excite them to the higher energy levels. Thus, when an atom is heated, the electrons excite to the higher energy level and then fall off to the ground state giving light.