  We analyze the production of the neutral Higgs particles of the Minimal
Supersymmetric extension of the Standard Model at the Fermilab Tevatron
collider. We consider the two main production and detection channels:
gluon--gluon and bottom quark fusion leading to Higgs bosons which subsequently
decay into tau leptons, $gg, b\bar b \to {\rm Higgs} \to \tau^+\tau^-$. We
update the production cross sections and the decay branching ratios and obtain
production rates that are significantly smaller at high masses than the ones
used by the CDF and D0 experiments in their search. We then evaluate the
various theoretical uncertainties that affect these rates, uncertainties that
have not been been considered in the experimental analyses and which turn out
to be rather large. Including these two effects will significantly loosen the
constraints obtained on the supersymmetric Higgs sector at the Tevatron.
