  Molecular hydrogen ions in weakly bound states close to the first
dissociation threshold are attractive quantum sensors for measuring the
proton-to-electron mass ratio and hyperfine-induced ortho-para mixing. The
experimental accuracy of previous spectroscopic studies relying on fast ion
beams could be improved by using state-of-the-art ion trap setups. With the
electric dipole moment vanishing in H$_2^+$ and preventing fast spontaneous
emission, radiative lifetimes of the order of weeks are found. We include the
effect of black-body radiation that can lead to photodissociation and
rovibronic state redistribution to obtain effective lifetimes for trapped ion
experiments. Rate coefficients for bound-bound and bound-continuum processes
were calculated using adiabatic nuclear wave functions and nonadiabatic
energies, including relativistic and radiative corrections. Effective lifetimes
for the weakly bound states were obtained by solving a rate equation model and
lifetimes in the range of 4 to 523~ms and $>$215~ms were found at room
temperature and liquid nitrogen temperature, respectively. Black-body induced
photodissociation was identified as the lifetime-limiting effect, which
guarantees the purity of state-selectively generated molecular ion ensembles.
The role of hyperfine-induced $g/u$-mixing, which allows pure rovibrational
transitions, was found to be negligible.
