What is the typical example $\pi \colon X \to B$ of a surjective morphism between topological (or $C^{\infty}$) manifolds, i.e. locally euclidean space via continuous bijections, where path lifting property fails to hold?
It seems for me that every morphism $\pi$ satisfies the path-lifting property.
On
There is a reason why "a lot" (rather, nice enough) of the maps you usually encounter actually do have the path-lifting property. Namely, a smooth submersion that is also proper is already a fiber bundle by Ehresmann's theorem and fiber bundles have the path-lifting property (even more generally, they have the homotopy lifting property).
However, this is of course not always the case. Consider, for exampe, the map $(-2\pi,2\pi)\rightarrow S^1,t\mapsto e^{it}$, which is even a local diffeomorphism. The path $[0,1]\rightarrow S^1,t\mapsto e^{2\pi it}$ does not lift through this map, as you can check. The point is that a lift would very much love to spread out from $0$ to $2\pi$, but the latter does not exist in the domain. This incompleteness is to say that the map isn't proper.
If by path-lifting property you mean that every continuous/smooth path $\gamma:[0,1]\to B$ has a lift, then then this need not be the case, even if we require that $\pi$ be a submersion or local diffeomorphism. As a counterexample, one can take a disjoint union of overlapping intervals, such as $$ \pi:(-\infty,1)\sqcup(-1,\infty)\to\mathbb{R} $$ where $\pi$ restricts to the inclusion map on each component.