We investigate the dynamical stability of a dark state of the atom-heteronuclear-trimer conversion system in a stimulated Raman adiabatic passage process. By casting the quantum Hamiltonian into an effective classical one and analyzing the eigenvalues of the Hamiltonian-Jacobi matrix obtained by linearizing the equations of motion around the fixed point corresponding to the dark state, we analytically obtain the condition for the occurrence of the dynamical instability of the atom-trimer dark state. Taking heteronuclear atoms of 87Rb and 41K as an example, we numerically give the unstable region. We find that the dynamical instability is induced by the interparticle interactions. Moreover，the occurrence of the dynamical instability depends on not only the emergence of the real or complex eigenvalues of the Hamiltonian-Jacobi matrix but also the scanning rate of the external fields.