No, even for $A=\mathbf{Z}$.

Take your favourite example of a complex of short exact sequences of abelian groups whose projective limit is not exact (see for example this math.stackexchange question).

Now just make a complex of long exact sequences by splicing together the short exact sequences a la

$$\to0\to A\to B\to C\to 0\to 0\to A\to B\to C\to 0\to\cdots$$

and you get failure of exactness of the projective limit all over the place, even though all the sequences in the complex are exact.

No, even for $A=\mathbf{Z}$.

Take your favourite example of a complex of short exact sequences of abelian groups whose projective limit is not exact (see for example this math.stackexchange question).

Now just make a complex of long exact sequences by splicing together the short exact sequences a la

$$\to0\to A\to B\to C\to 0\to 0\to A\to B\to C\to 0\to\cdots$$

and you get failure of exactness of the projective limit all over the place, even though all the sequences in the complex are exact.