No, something on Earth cannot be so heavy that it causes the entire planet to move. The movement of an entire planet is influenced by gravitational forces acting on a massive scale, such as the interactions between Earth and other celestial bodies like the Moon and the Sun. The motion of Earth itself is primarily determined by the distribution of mass within the planet and external factors like these gravitational forces.
On a smaller scale, individual objects can certainly have an impact on their immediate surroundings through their mass and gravitational attraction. For example, large buildings, mountains, or even moving masses like glaciers can exert local gravitational forces that affect the Earth's crust and cause tectonic movements. However, these effects are localized and limited in scope, not capable of moving the entire planet.