The Visual-Inertial Simultaneous Localization and Mapping (VI-SLAM) algorithms which are mostly based on static assumption are widely used in fields such as robotics, UAVs, VR, and autonomous driving. To overcome the localization risks caused by dynamic landmarks in most VI-SLAM systems, a robust visual-inertial motion prior SLAM system, named IDY-VINS, is proposed in this paper which effectively handles dynamic landmarks using inertial motion prior for dynamic environments to varying degrees. Specifically, potential dynamic landmarks are preprocessed during the feature tracking phase by the probabilistic model of landmarks' minimum projection errors which are obtained from inertial motion prior and epipolar constraint. Subsequently, a robust and self-adaptive bundle adjustment residual is proposed considering the minimum projection error prior for dynamic candidate landmarks. This residual is integrated into a sliding window based nonlinear optimization process to estimate camera poses, IMU states and landmark positions while minimizing the impact of dynamic candidate landmarks that deviate from the motion prior. Finally, a clean point cloud map without `ghosting effect' is obtained that contains only static landmarks. Experimental results demonstrate that our proposed system outperforms state-of-the-art methods in terms of localization accuracy and time cost by robustly mitigating the influence of dynamic landmarks.