I am going to concentrate on the underlying categorical issue here and leave the homological algebra to the experts in that subject.
A direct limit is also known as a colimit. Yours, I take it, is over a sequence $N$, and as such is called directed or filtered.
Being an short exact sequence $0\to A\to B\to C\to 0$ amounts three things:
$A\to B$ is a monomorphism or $0$ is its kernel, which are kinds of limit (or projective limit in old terminology), but finitary ones;
$B\to C$ is an epimorphism, or $C$ is its image, which are other kinds of *colimit property, this time finitary ones.
the image of $A\to B$ is the kernel of $B\to C$, which combines properties of both kinds.
Now, limits commute with limits and colimits commute with colimits.
The question is whether filtered colimits commute with finite limits.
At least, I presume that is what a Grothendieck category might mean.