Your first screenshot of the 3D Viewport is not the scene you have rendered. It's not the plate hiding the cup, the render doesn't even show the plate - it's below the plane. In your file you have moved the plane upwards so it's halfway in the cup - that's why everything is hidden below.
EDIT: If you only move the plate with the cup (parented to the plate) upwards instead of the plane downwards, they will be back under the plane as soon as you render or play an animation, because the plate has keyframes on location, rotation and scale which place it below the plane.
Your 3D Viewport when I open your file:

Side view in Wireframe mode:
