Any coCartesian fibration which is a localization is a universal localization, since coCartesian fibrations are closed under pullbacks, and the fibers are stable under pullbacks as well. 

In particular, any localization which inverts all maps, since it has an $\infty$-groupoid as target, is a coCartesian fibration and is therefore a universal localization. 

Any (co)reflexive localization is also universal because (co)reflexive localizations are closed under pullbacks. 

So we have to look a bit further for examples of non-universal localizations. An easy thing to come up with an example is to observe that $C[W^{-1}]$ typically has more morphisms than $C$, so the pullback $\Delta^1\times_{C[W^{-1}]}C$ is likely to be too small. Let $p:C\to C[W^{-1}]$ denote the localization functor. 

Indeed, given a map $f: x\to y$ in $C[W^{-1}]$, objects of $\Delta^1\times_{C[W^{-1}]} C$ are objects of the fibers $p^{-1}(x), p^{-1}(y)$, there are no morphisms in the direction $y\to x$, and morphisms lying over $0\to 1$ are maps from some $x_0\in p^{-1}(x)$ to some $y_0\in p^{-1}(y)$ lifting $f$. 

Consider for example the following : $C$ is $a\to b, a\to c, d\to c$ and nothing else. If you invert $a\to c$, you can prove that the resulting category is $[2]= 0\leq 1\leq 2$ with $p: d\mapsto 0, a,c\mapsto 1, b\mapsto 2$, so that there are no maps from the fiber over $0$ to the fiber over $2$. Thus in this example (which I recommend drawing out), $\Delta^1\times_{C[W^{-1}]}C$ has two objects and no maps in either direction, and so it cannot localize onto $\Delta^1$.