The induced morphism of Hodge structures for any map of varieties will be an isomorphism if and only if the induced map on cohomology (forgetting the Hodge structure) is an isomorphism. This follows from the fact that morphisms of mixed Hode structures are strict with repsect to both the Hodge and weight filtration; see Deligne, Theorie de Hodge II.
In your setup, if the map of underlying toplogical spaces is a homeomorphism it induces an isomorphism on cohomology and so by the above also an isomorphism of mixed Hodge structures. This might not always happen: consider for example the map $\mathbb{A}^1 - \{0\} \ \sqcup \{0\} \to \mathbb{A}^1$. Or consider the normalisation of a nodal curve with one of the points lying above the node removed.