  A comprehensive study of the effect of an unstable tau neutrino on primordial
nucleosynthesis is presented. The standard code for nucleosynthesis is modified
to allow for a massive decaying tau neutrino whose daughter products include
neutrinos, photons, $e^\pm$ pairs, and/or noninteracting (sterile) daughter
products. Tau-neutrino decays influence primordial nucleosynthesis in three
distinct ways: (i) the energy density of the decaying tau neutrino and its
daughter products affect the expansion rate tending to increase $^4$He, D, and
$^3$He production; (ii) electromagnetic (EM) decay products heat the EM plasma
and dilute the baryon-to-photon ratio tending to decrease $^4$He production and
increase D and $^3$He production; and (iii) electron neutrinos and
antineutrinos produced by tau-neutrino decays increase the weak rates that
govern the neutron-to-proton ratio, leading to decreased $^4$He production for
short lifetimes ($\la 30\sec$) and masses less than about $10\MeV$ and
increased $^4$He production for long lifetimes or large masses. The precise
effect of a decaying tau neutrino on the yields of primordial nucleosynthesis
and the mass-lifetime limits that follow depend crucially upon decay mode. We
identify four generic decay modes that serve to bracket the wider range of
possibilities:
