Fusing low-resolution (LR) hyperspectral images (HSIs) with high-resolution (HR) multispectral images (MSIs) is a significant technology to enhance the resolution of HSIs. Despite the encouraging results from deep learning (DL) in HSI-MSI fusion, there are still some issues. First, the HSI is a multidimensional signal, and the representability of current DL networks for multidimensional features has not been thoroughly investigated. Second, most DL HSI-MSI fusion networks need HR HSI ground truth for training, but it is often unavailable in reality. In this study, we integrate tensor theory with DL and propose an unsupervised deep tensor network (UDTN) for HSI-MSI fusion. We first propose a tensor filtering layer prototype and further build a coupled tensor filtering module. It jointly represents the LR HSI and HR MSI as several features revealing the principal components of spectral and spatial modes and a sharing code tensor describing the interaction among different modes. Specifically, the features on different modes are represented by the learnable filters of tensor filtering layers, the sharing code tensor is learned by a projection module, in which a co-attention is proposed to encode the LR HSI and HR MSI and then project them onto the sharing code tensor. The coupled tensor filtering module and projection module are jointly trained from the LR HSI and HR MSI in an unsupervised and end-to-end way. The latent HR HSI is inferred with the sharing code tensor, the features on spatial modes of HR MSIs, and the spectral mode of LR HSIs. Experiments on simulated and real remote-sensing datasets demonstrate the effectiveness of the proposed method.