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.
Indeed, they do if the category described is by a finitary algebraic theory.
I presume that a Grothendieck category is like this, though Zhen Lin says otherwise.
As David White says, the interest in filtered colimits arises from the fact that
they are the ones that commute with finitary stuff such as limits in $\bf Set$.
However, this is not a theorem of categories in general.
As Zhen Lin points out, ${\bf Set}^{op}$ does not have this property.
Limits and colimits in $\bf Set$ behave quite differently, as I emphasise in
Chapter V of my book,
Practical Foundations of Mathematics.
A preorder in which this happens is called meet-continuous. See Counterexamples O 4.5 of A Compendium of Continuous Lattices for complete lattices that are not meet-continuous.
(I had difficulty with my Internet connection when I originally posted this, so some of the intended text got deleted by mistake.)
I repeat that I am not an expert on homological algebra, but I note that the
Wikipedia article on Grothendieck categories says explicitly that "direct limits (a.k.a. filtered colimits) of exact sequences are exact".
Best Answer
$\DeclareMathOperator\Hom{Hom}$The answer seems to be positive.
Let $\psi_i: X_i \rightarrow Y_i$. Consider the directed system of exact sequences
$$ 0 \longrightarrow K_i \longrightarrow X_i\otimes- \xrightarrow{\psi_i \otimes -} Y_i \otimes -$$
in the functor category. Its direct limit equals an exact sequence
$$ 0 \longrightarrow K \longrightarrow X\otimes- \xrightarrow{\psi \otimes -} Y \otimes -$$
since direct limits are exact and tensors commute with them. Now $M$ is pure-injective, so the functor $M\otimes-$ is injective and there exists an inverse system of short exact sequences
$$ \Hom(Y_i\otimes-,M\otimes-) \xrightarrow{\Hom(\psi _i \otimes - , M\otimes -)} \Hom(X_i\otimes-, M\otimes-)\longrightarrow \Hom(K_i, M \otimes-)\longrightarrow 0. $$
The natural isomorphism $\Hom(\psi _i \otimes - , M\otimes -)\cong \Hom(\psi_i, M)$ implies $\Hom(K_i, M \otimes-) = 0$. Because $\Hom(-,-)$ turns direct limits in the first argument into inverse limits, taking the inverse limit of the inverse system yields
$$ \Hom(Y\otimes-,M\otimes-) \xrightarrow{\Hom(\psi \otimes - , M\otimes -)} \Hom(X\otimes-, M\otimes-)\longrightarrow \Hom(K, M \otimes-) = 0.$$
Hence $\Hom(\psi, M)\cong \Hom(\psi \otimes - , M\otimes -) $ is surjective.