I have a possible answer for this question finally, but I am not that sure.
First of all, the question is not that 'correct' because the sequence $Z[T]\stackrel{\cdot T}{\longrightarrow} Z[T]\rightarrow Z[T]/(T)$ does not lie in the category $CRing$ but in $Mod$, neither does $A\rightarrow A\rightarrow \displaystyle A/\!\!/f$, which actually lies in infinity category $Mod(Z[T])$. S0 we should consider everything here in the stable $\infty$ category $Mod$.
So in $Mod(Z[T])$, by tensor product with $A\otimes_{Z[T]}^{L}(-)$, we have again an exact sequence $A\otimes_{Z[T]}^{L} Z[T]\rightarrow A\otimes_{Z[T]}^{L} Z[T]\rightarrow A\otimes_{Z[T]}^{L} Z[T]/(T)$, and it is fiber and also cofiber sequence, and recall that what we need is actually $\textbf{a fiber sequence of underlying spaces}$ instead of some fiber sequence in $\textbf{SCRing}$, luckily, the infinite loop functor $\Omega^{\infty}(-)$ functor preserve all small limits, thus we get a fiber sequence in $\textbf{Spc}$.