Among the growing list of confirmed exoplanets, the number of planets identified in dense star clusters remains sparse. Previous analyses have suggested that this may be due in part to dynamical interactions that can unbind planets from their host stars, limiting the survival of planetary systems in clusters. Thus, alternative detection strategies may be necessary to study planets in clusters that may no longer be bound to a host. Here, we use N-body models to explore the evolution of planetary systems in dense star clusters. Depending on various initial conditions, we show that 10%-50% of primordial planetary systems are broken through dynamical encounters over a cluster's full lifetime, populating clusters with "free-floating" planets. Furthermore, a large number (30%-80%) of planets are ejected from their host cluster through strong dynamical encounters and/or tidal loss. Additionally, we show that planets naturally mix with stellar-mass black holes (BHs) in the cores of their host cluster. As a consequence, up to a few hundred planets will be tidally disrupted through close passages of BHs. We show that these BH-planet tidal disruption events (TDEs) occur in clusters at a rate of up to 10^-5 yr^-1 in a Milky-Way-type galaxy. In principle, these BH-planet TDEs may be detected by upcoming transient surveys such as the Large Synoptic Survey Telescope at a rate of a few events per year, although identification of these events may prove challenging. The observed rate of BH-planet TDEs could place new constraints upon the formation and survival of planetary systems and BHs in dense star clusters.