Accurate interpolation algorithms are highly desired in various theoretical and engineering scenarios. Unlike the traditional numerical algorithms that have exact zero-residual constraints on observed points, the neural network-based interpolation methods exhibit non-zero residuals at these points. These residuals, which provide observations of an underlying residual function, can guide predicting interpolation functions, but have not been exploited by the existing approaches. To fill this gap, we propose Hierarchical INTerpolation Network (HINT), which utilizes the residuals on observed points to guide target function estimation in a hierarchical fashion. HINT consists of several sequentially arranged lightweight interpolation blocks. The first interpolation block estimates the main component of the target function, while subsequent blocks predict the residual components using observed points residuals of the preceding blocks. The main component and residual components are accumulated to form the final interpolation results. Furthermore, under the assumption that finer residual prediction requires a more focused attention range on observed points, we utilize hierarchical local constraints in correlation modeling between observed and target points. Extensive experiments demonstrate that HINT outperforms existing interpolation algorithms significantly in terms of interpolation accuracy across a wide variety of datasets, which underscores its potential for practical scenarios.