The employment of partial zero-forcing (PZF) receivers at the base stations represents an efficient and low-complexity technique for uplink interference management in cellular networks. In this paper, we focus on the performance analysis of ultra-dense networks (UDNs) in which the multi-antenna receivers adopt PZF. We provide both integral expressions and tight closed-form approximations for the probability of successful transmission, which can be used to accurately evaluate the optimal tradeoff between interference cancellation and array gain. Numerical results show that no more than half of the available degrees of freedom should be used for interference cancellation.