Number of induced paths in an interval graph
Consider the following example:



Say the total number of intervals is $n$. To get from one red extremity to the other, you can take a path which alternates between green and blue intervals. Each of those paths is induced, and there are $2^\frac{n-2}{2}$ such paths. Thus, the number of induced paths between two vertices in an interval graph is not necessarily polynomial in the number of vertices.