This paper analyzes the impact of user mobility in multi-tier heterogeneous networks. We begin by obtaining the handoff rate for a mobile user in an irregular cellular network with the access point locations modeled as a homogeneous Poisson point process. The received signal-to-interference-ratio (SIR) distribution along with a chosen SIR threshold is then used to obtain the probability of coverage. To capture potential connection failures due to mobility, we assume that a fraction of handoffs result in such failures. Considering a multi-tier network with orthogonal spectrum allocation among tiers and the maximum biased average received power as the tier association metric, we derive the probability of coverage for two cases: 1) the user is stationary (i.e., handoffs do not occur, or the system is not sensitive to handoffs); 2) the user is mobile, and the system is sensitive to handoffs. We derive the optimal bias factors to maximize the coverage. We show that when the user is mobile, and the network is sensitive to handoffs, both the optimum tier association and the probability of coverage depend on the user's speed; a speed-dependent bias factor can then adjust the tier association to effectively improve the coverage, and hence system performance, in a fully-loaded network.