We demonstrate the existence of the first-class constraints on the massless Abelian 3-form theory which generate the classical gauge symmetry transformations for this theory in any arbitrary D-dimension of spacetime. We write down the explicit expression for the generator in terms of these first-class constraints. Using the celebrated Noether theorem, corresponding to the gauge symmetry transformations, we derive the Noether conserved current and conserved charge. The latter is connected with the first-class constraints of the theory in a subtle manner as we demonstrate clearly in our present investigation. We comment on the first-class constraints within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the conserved (anti-)BRST charges are the generalizations of the above generator for the classical gauge symmetry transformation. The standard Noether conserved (anti-)BRST charges are found to be non-nilpotent. We derive the nilpotent versions of the (anti-)BRST charges. One of the interesting observations of our present endeavor is the result that only the nilpotent versions of the conserved (anti-)BRST charges lead to the annihilation of the physical states by the operator form of the first-class constraints at the quantum level which is consistent with the Dirac quantization condition for the systems that are endowed with any kind of constraints. We comment on the existence of the Curci-Ferrari (CF) type restrictions from different theoretical angles, too. •Classical local gauge symmetry of the D-dimensional massless Abelian 3-form theory is elevated to its quantum counterparts (anti-)BRST symmetries.•Noether (anti-)BRST charges are found to be non-nilpotent.•Nilpotent versions of the (anti-)BRST charges are derived.•Physicality criteria with the nilpotent (anti-)BRST charges lead to the annihilation of the physical states by the operator form of the first-class constraints.