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.