Flexible intelligent metasurfaces (FIMs) offer a new solution for wireless communications by introducing morphological degrees of freedom, dynamically morphing their three-dimensional shape to ensure multipath signals interfere constructively. However, realizing the desired performance gains in FIM systems is critically dependent on acquiring accurate channel state information across a continuous and high-dimensional deformation space. Therefore, this paper investigates this fundamental channel estimation problem for FIM assisted millimeter-wave communication systems. First, we develop model-based frameworks that structure the problem as either function approximation using interpolation and kernel methods or as a sparse signal recovery problem that leverages the inherent angular sparsity of millimeter-wave channels. To further advance the estimation capability beyond explicit assumptions in model-based channel estimation frameworks, we propose a deep learning-based framework using a Fourier neural operator (FNO). By parameterizing a global convolution operator in the Fourier domain, we design an efficient FNO architecture to learn the continuous operator that maps FIM shapes to channel responses with mesh-independent properties. Furthermore, we exploit a hierarchical FNO (H-FNO) architecture to efficiently capture the multi-scale features across a hierarchy of spatial resolutions. Numerical results demonstrate that the proposed H-FNO significantly outperforms the model-based benchmarks in estimation accuracy and pilot efficiency. In particular, the interpretability analysis show that the proposed H-FNO learns an anisotropic spatial filter adapted to the physical geometry of FIM and is capable of accurately reconstructing the non-linear channel response across the continuous deformation space.