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 $Mod$ (This is obvious because the first arrow is not a ring morphism), neither does $A\rightarrow A\rightarrow \displaystyle A/\!\!/f$, which actually lies in infinity category $Mod(Z[T])$. So we should consider everything here in the stable $\infty$ category $Mod$.
In $Mod(Z[T])$, by tensor product with $A\otimes_{Z[T]}^{L}(-)$, we have again an exact sequence $A\otimes_{Z[T]}^{L} Z[T]\rightarrow A\otimes_{Z[T]}^{L} Z[T]\rightarrow A\otimes_{Z[T]}^{L} Z[T]/(T)$, it is both fiber and also cofiber sequence because we are in stable infinity category, 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 preserves all small limits, thus we get a fiber sequence in $\textbf{Spc}$.
It's worth mentioning that the derived tensor product indeed is constructed from a pushout diagram in $\textbf{SCRing}$, but it is not a pushout when we consider the underlying module structure in $\textbf{Mod}$. To see this, just take the category of modules over some filed $k$, then pushout in the category of vector space is exactly the cartesian product whose dimension is different from tensor product. Instead, 'tensor product $\otimes$' usually inherits from some more basic underlying category, for example here in the $E_{\infty}$ context, the tensor product(or the symmetric monoidal structure) actually comes from the smash product in the infinity category of spectra $\textbf{Sp}$. So this is not a question about commutativity of limits and colimits because in $Mod$, tensor product is not a colimit at all.
More precisely, to fit in the definition we used at the beginning $\displaystyle A/\!\!/f=A\otimes_{Z[T]}Z[T]/(T)$ which is the pushout in $SCRing$, for all these 'rings', consider their underlying 'modules' and this pushout has the tensor product of 'modules' as its underlying 'module', thus they have the same underlying spectrum in $\textbf{Sp}$. This explains our abuse of notation of tensor product $\otimes$ here. Also see this relevant post.