We address the user association problem in multi-tier in-band full-duplex (FD) networks. Specifically, we consider the case of decoupled user association (DUA) in which users (UEs) are not necessarily served by the same base station (BS) for uplink (UL) and downlink (DL) transmissions. Instead, UEs can simultaneously associate to different BSs based on two independent weighted path-loss user association criteria for UL and DL. We use stochastic geometry to develop a comprehensive modeling framework for the proposed system model where BSs and UEs are spatially distributed according to independent point processes. We derive closed-form expressions for the mean rate utility in FD, half-duplex (HD) DL, and HD UL networks as well as the mean rate utility of legacy nodes with only HD capabilities in a multi-tier FD network. We formulate and solve an optimization problem that aims at maximizing the mean rate utility of the FD network by optimizing the DL and UL user association criteria. We investigate the effects of different network parameters including the spatial density of BSs and power control parameter. We also investigate the effect of imperfect self-interference cancellation (SIC) and show that it is more severe at UL, where there exist minimum required SIC capabilities for BSs and UEs for which FD networks are preferable to HD networks; otherwise, HD networks are preferable. In addition, we discuss several special cases and provide guidelines on the possible extensions of the proposed framework. We conclude that DUA outperforms coupled user association (CUA) in which UEs associate to the same BS for both UL and DL transmissions.