  For the St$\ddot u$ckelberg-modified massive Abelian 3-form theory in any
arbitrary D-dimension of spacetime, we show that its classical gauge symmetry
transformations are generated by the first-class constraints. We establish that
the Noether conserved charge (corresponding to the local gauge symmetry
transformations) is same as the standard form of the generator for the
underlying local gauge symmetry transformations (expressed in terms of the
first-class constraints). We promote these classical local, continuous and
infinitesimal gauge symmetry transformations to their quantum counterparts
Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations which
are respected by the coupled (but equivalent) Lagrangian densities. We derive
the conserved (anti-)BRST charges by exploiting the theoretical potential of
Noether's theorem. However, these charges turn our to be non-nilpotent. Some of
the highlights of our present investigation are (i) the derivation of the
off-shell nilpotent versions of the (anti-)BRST charges from the standard
non-nilpotent Noether conserved (anti-)BRST charges, (ii) the appearance of the
operator forms of the first-class constraints at the quantum level through the
physicality criteria w.r.t. the nilpotent versions of the (anti-)BRST charges,
and (iii) the deduction of the CF-type restrictions from the straightforward
equality of the coupled (anti-)BRST invariant Lagrangian densities as well as
from the requirement of the absolute anticommutativity of the off-shell
nilpotent versions of the (anti-)BRST charges.
