Suppose, for each $n>0$, I have two complexes of abelian groups $(A\_{n}, d)$ and $(B\_{n},d')$
and a quasi-isomorphism 
$$f\_{n}: A\_{n} \rightarrow B\_{n}.$$
Furthermore, suppose I have maps of complexes $A\_1 \rightarrow A\_2 \rightarrow A\_3 \dots$
and similarly for $B\_n$, compatible with $f\_n$.


Is it true that the map of direct limit complexes
$$\lim\_{n} f\_{n}: \lim A \rightarrow \lim B$$
is also a quasi-isomorphism?


Complexes can mean unbounded or maybe just bounded above/below, depending on your preference.


Also: what if I replace the limit over natural numbers with a more general filtered colimit?



Direct limits and quasi-isomorphism
Yes indeed. You may assume that the complexes belong to a Grothendieck category if you wish, or just stick with abelian groups. (Grothendieck category: abelian category with exact filtered direct limits that possesses a generator). In the countable case, you have Milnor's exact sequence


$$
\oplus\_{n} A\_n \longrightarrow \oplus\_{n} A\_n \longrightarrow \lim\_{n} A\_n
$$


The first arrow, usually denoted by $1-shift$ takes an element in $A\_n$ and sends it to itself in $A\_n$ and to the negative of its image by the transition map in $A\_{n+1}$. It is an exercise that $1-shift$ is injective on abelian groups (and the proof can be adapted to the general situation).


You have a similar sequence for the $B\_n$'s and a map of exact sequences given by $\oplus\_{n} f\_n$ in the first and second factor and the induced map in the third. When you interpret this diagram in the derived catdegory it is clear that $\oplus\_{n} f\_n$ is a quasi-isomorphism and your diagram of exact sequences becomes a diagram of triangles. This implies that the induced map is a quasi-isomorphism.


For certain more general systems there are some results in my joint paper with Jeremías & Souto, [here](http://webspersoais.usc.es/export/sites/default/persoais/leo.alonso/papeles/locores4c.pdf) or in the [CJM](http://cms.math.ca/cjm/v52/p225/) site. You can find here an exposition of homotopy colimitis in the context of complexes. Hope this helps.


