I was thinking about whether this positive result would hold in the category of locally convex spaces also...
Here is what I got so far: The direct limit of a locally convex system consists of the final topology on the direct limit of the underlying linear system. (I tried to prove this formally here.) So the direct limit of the vector spaces is $X:=\oplus_\alpha X_\alpha/\sim$ where $\sim$ is the subspace generated by the all elements of the form $x-f_{\alpha\beta}(x)$.
Because we deal with directed posets there is another, nicer representation of $X$, namely the set $\sqcup_\alpha X_\alpha/\sim$ where $\sqcup$ denotes disjoint union and $x\sim y$ for $x\in X_\alpha,y\in X_\beta$ iff there exists some $\alpha,\beta\preceq\gamma$ such that $f_{\alpha\gamma}(x)=f_{\beta\gamma}(y)$ (compare again with Stefan's brilliant answer here). Addition can then always be carried out in a common domain, i.e. $[x]_\sim+[y]_\sim:=[f_{\alpha\gamma}(x)+f_{\beta\gamma}(y)]_\sim$. I will denote with $X$ only this alternative construction from here on.
So the embedding $g_\alpha:X_\alpha\rightarrow X$ is closed on its image iff for any closed $A\subseteq X_\alpha$ and $x\in X_\alpha\setminus A$ there is some neighbourhood $U\in{\cal U}(x)$ in $X$ such that $U\cap A=\emptyset$, i.e. $X_\alpha\cap\overline{g_\alpha(A)}=A$.
But that means for $\alpha\preceq\beta$ we have to find open, convex neighbourhoods $U_\beta\in{\cal U}(f_{\alpha\beta}(x))$ in $X_\beta$ which do not intersect with $f_{\alpha\beta}(x)$ and which are consistent w.r.t. the $f_{\beta\gamma}$, i.e. $f_{\beta\gamma}^{-1}(U_\gamma)=U_\beta$ whenever $\alpha\preceq\beta\preceq\gamma$. This is exactly the condition for an induced basic neighbourhood in $X$. (For $\alpha\npreceq\beta$ we can choose the $U_\beta$ by passing down from some $\alpha,\beta\preceq\gamma$.)
Now here is where I'm kind of stuck, it seems to me like there is no easy way out and countexamples could be found. I asked myself this more general question with the problem of finding a consistent choice of neighbourhoods in mind and found a counterexample but I'm not sure if I can transform it so to also disprove this question.