This complex (in simultaneously a more general setting, where you have several elements $t_1,\ldots,t_p$, and a more special setting, because only the case of $R$ being a free algebra was studied then) was introduced by Shafarevich [E. S. Golod, I. R. Shafarevich, “On the class field tower”, Izv. Akad. Nauk SSSR Ser. Mat., 28:2 (1964), 261–272], and was studied somewhat thoroughly recently, see, e.g. [E. S. Golod, “Homology of the Shafarevich complex and noncommutative complete intersections”, Fundam. Prikl. Mat., 5:1 (1999), 85–95] - this paper again dealing with the case of free $R$. As the latter paper suggests, this complex is a noncommutative generalisation of the Koszul complex in the commutative case, detecting the "noncommutative complete intersections" usually known as "strongly free sets" from a paper of Anick [D. J. Anick, Non-commutative graded algebras and their Hilbert series, J. Algebra, Volume 78, Issue 1, September 1982, Pages 120-140].
Finally, in the case of arbitrary (graded) $R$ this complex is discussed in a paper of Piontkovski [http://arxiv.org/abs/math/0606279] for the purpose of studying "relative noncommutative complete intersections". You would be especially interested in "Theorem-Definition 2.3" from that paper.
(For those who cannot be bothered to check the references, one condition of Piontkovski's paper is completely analogous to the complete intersection result stating that $A=B[f_1,\ldots,f_k]$ for a regular sequence $f_1,\ldots,f_k$: the complex in question is acyclic iff $R=(R/r)\langle r\rangle$. Here one has to make some assumptions, e.g. assume $R$ graded and $r$ being homogeneous of positive degree, so this does not cover central units mentioned in the question, whereas Tyler Lawson's example is the simplest instance of this result.)