Let me first explain the statement of the question and then give some indication why the answer might be 'yes'. By a space I mean, say, a simplicial set and by rational I mean rational in the sense of Bousfield, i.e. local with respect to the homology theory $H\mathbb{Q}$. If we denote the category of spaces by $\mathcal{S}$ then the category of rational spaces $\mathcal{S}_\mathbb{Q} \subseteq \mathcal{S}$ is a full subcategory and the question is whether it is closed under filtered colimits. This would in particular imply that the $\infty$-category of rational spaces is compactly generated. Note that it is known that this $\infty$-category is generated by $\kappa$-compact objects for $\kappa$ the successor cardinal of $\omega$ (by results of Bousfield). But whether it is compactly generated is unclear to me and this is really what I want to know, even if the inclusion $\mathcal{S}_\mathbb{Q} \subseteq \mathcal{S}$ is not necessarily $\omega$-continuous.

Now let me give some indication why the answer could be 'yes': first a simply connected (or more generally nilpotent) space is rational if and only if its homotopy groups are uniquely divisible. This property is obviously closed under filtered colimits. Thus a filtered colimit of simply connected, rational spaces is again rational. Bousfield characterizes rationality for arbitrary spaces as a property of homotopy groups, but this property is also homotopy theoretic in nature so that I am unable to decide if it is preserved by filtered colimits.

The second indication is that one can study the stable analogue of my question: is a filtered colimit of rational spectra again rational. The answer is yes. This is in fact equivalent to the fact that rationalization is a smashing localization. This observation also shows that the class of $\mathbb{S}/p$-local spectra for a prime $p$ cannot be closed under filtered colimits since $p$-completion is not smashing. A consequence is that $\mathbb{F}_p$-local spaces (a.k.a. $\mathbb{S}/p = M(\mathbb{Z}/p)$-local spaces) are not closed under filtered colimits.