Unless I am severely mistaken, the answer to the question as stated is "no". (There is by the way some slight disagreement in the literature as to what "filtered category" means; for many, a filtered poset means a nonempty poset where any two elements have an upper bound, but in Sheaves in Geometry and Logic, Mac Lane and Moerdijk mean that any two elements have a lower bound. I don't think this affects my answer.)

Let us take for instance the ring of $p$-adic integers as inverse limit of the (directed in either sense) system

$$\ldots \to \mathbb{Z}/(p^{n+1}) \to \mathbb{Z}/(p^n) \to \ldots$$

Each of these commutative rings $A$ can be viewed as an $Ab$-enriched category, and the abelian category of left modules over $A$ can be viewed as the $Ab$-enriched category of $Ab$-enriched functors $A \to Ab$. So we are homming into $Ab$, and the usual expectation is that a contravariant hom-functor takes colimits to limits, but not limits to colimits; the question as stated has to do with the latter situation.

The filtered colimit (and it would be reasonable to interpret "colimit" in a 2-categorical sense) of the system of full subcategory inclusions

$$\ldots \to Ab^{\mathbb{Z}/(p^n)} \to Ab^{\mathbb{Z}/(p^{n+1})} \to \ldots$$

can be taken to be the union of these (abelian) categories. The union is just the category of those abelian groups where each object is completely annihilated by some $p^n$ (not uniformly of course; the $n$ depends on the group); in particular, they are all torsion. On the other hand, most objects in the category of modules over the $p$-adic integers, for example the $p$-adic integers itself, are not torsion. So there's your counterexample.

But if we turn the directions around, then indeed colimits are taken to limits. There are some slightly tricky issues to deal with, because "morally" we really ought to be working with 2-categorical limits -- weak limits to be precise. To do it right, we should be working with the category of commutative rings as a 2-category (construing ring homomorphisms as enriched functors and taking 2-cells as natural transformations), but if I'm not mistaken, weak colimits in this 2-category of commutative rings coincide with ordinary colimits of commutative rings, and we can relax. Anyway, it is a general feature that homming into an object like $Ab$, giving here a functor

$$Ab^{-}: \text{CRing}^{op} \to \text{AbCat},$$

takes (weak) colimits to (weak) limits.

limitsin the category of commutative rings are taken tocolimitsin the (2-)category of abelian categories? Or did you actually mean to ask whether (filtered) colimits in commutative rings are taken to limits in the category of abelian categories? I believe the answer is "no" for the first and "yes" for the second (under reasonable interpretations, which depend on the exact interpretation of the category or 2-category of abelian categories), and I think I'm prepared to say more, but I need to ask first whether you're sure this is what you meant to ask. – Todd Trimble♦ Apr 9 '11 at 11:22