examples - Request: A Serre fibration that is not a Dold fibration

You've already answered your own question, but here is another example.

Let $f: mathbb{Q}^delta to mathbb{Q}$ be the obvious map from the rational numbers with the discrete topology to the rational numbers with the usual topology. Let $M_f$ be the mapping cylinder. Then the projection
$$p:M_f to [0,1]$$
is a Serre fibration. This follows because any map of a disc into $mathbb{Q}$ factors through f. Hence as far as discs are concerned, $M_f$ might as well be $mathbb{Q}^delta times [0,1]$.

However this projection is not a Dold fibration. It is easy to construct a diagram using $Y = mathbb{Q}$ which will have no weak homotopy lift. Indeed consider the projection map $mathbb{Q} times [0,1] to [0,1]$ with the obvious initial lift. Any other initial lift vertically homotopic to this one in fact coincides with this one, hence it is easy to see that there is no weak lift of this map.

By replacing $mathbb{Q}^delta$ and $mathbb{Q}$ with their cones, we get a similar example where now the base, total space, and fibers are contractible. Hence they are path-connected and also have the homotopy type of CW complexes. So this also answers Ronnie Browns question (but surely the answer to that has been known for some time).