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Ultra-dense multi-tier cellular networks have recently drawn the attention of researchers due to their potential efficiency in dealing with high-data rate demands in upcoming 5G cellular networks. These networks consist of multi-tier base stations including micro base stations with very high-system capacity and short inter-site distances, overlooked by central macro base stations. In this way, network densification is achieved in the same area as that of traditional mobile networks, which offers much higher system capacity and bandwidth reuse. This paper utilizes a well-known analytical tool, stochastic geometry for modeling and analyzing interference in ultra-dense multi-tier cellular networks. Primarily, we have studied different factors affecting the system capacity including the network densification, cell load, and multi-tier interference. The role of the ergodic channel capacity is also discussed. Moreover, the effects of channel interference, system bandwidth, and the network densification on the spectral and energy efficiencies of the network are observed. Finally, the results show that the network densification and the cell load have a profound impact on system performance as well as spectral and energy efficiencies of the networks.