Cesium lead iodide, CsPbI₃, is an optoelectronic material of large interest for various technological applications; however, fundamental questions surrounding the vibrational dynamics of this material, especially regarding its role in structural phase transitions, remain to be elucidated. Here, in a combined variable temperature inelastic neutron scattering (INS) and machine-learning based molecular dynamics (MD) simulation study, we show that the stable phase at room temperature, i.e., the nonperovskite δ-phase, exhibits phonon modes with weak anharmonicity with only a weak temperature dependence from 10 K all the way up to the transition to the cubic perovskite α-phase at approximately 600 K. In contrast, the α-phase features anharmonic and damped vibrational dynamics, mainly associated with overdamped tilting motions of the PbI₆ octahedra. Crucially, these overdamped tilting modes, which relate to the tetragonal and orthorhombic distorted perovskite phases (β- and γ-phase, respectively) formed at lower temperatures, stay overdamped by more than 100 K above the respective phase transition. This suggests a flat energy landscape of octahedral tilting motions in α-CsPbI₃ and with structural fluctuations on the picosecond time scale with tilting patterns that locally resemble the structure of the β- and γ-phases. The vibrational dynamics of α-CsPbI₃ are also characterized by pronounced anharmonic motions with large thermal displacements of the Cs⁺ ions, but these modes remain underdamped at 600 K.