Suppose that I have a morphism of schemes of finite type over $\mathbb{C}$, which is a bijection on closed point (but in general I don't know if there exists an inverse for it).


I would like to know if there exist sufficient conditions on the source or the target such that the induced morphism of Hodge structures is an isomorphism.



When does a bijective morphism of schemes induce an isomorphism of Hodge structures?
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.


