Every fibration with totally path-disconnected fibers has the unique path lifting property (2.2.5 of E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966) implying that $p_{\ast}$ is injective. Surjectivity is the main difficulty but the overall approach is similar to that for covering maps. In particular, for every based map $f:X\to Y$, there is a unique based function $\tilde{f}:X\to\widetilde{Y}$ such that $p\circ\tilde{f}=f$ (defined using the fact that every path has a unique lift). The difficulty is knowing when $\tilde{f}$ is continuous. The answer is likely to be no in general but I can't think of an explicit counterexample at the moment.

If you are willing to allow $X$ to be simply connected and "$\Delta$-generated" (the quotient of a topological sum of simplices) then the answer is yes. To prove the continuity of $\tilde{f}$ you factor through the $\Delta$-generated coreflection of $\widetilde{Y}$. Thus if $X$ is a CW-complex, then you have an affirmative answer. If $X$ is simply connected, locally path-connected and first countable, then the answer is also affirmative.