Yes, the center of the Earth does experience gravitational time dilation, despite the absence of a net gravitational force. Gravitational time dilation occurs due to the presence of a gravitational field, which affects the passage of time.
According to Einstein's theory of general relativity, gravitational time dilation arises from the curvature of spacetime caused by massive objects. The greater the gravitational field, the more spacetime is curved, resulting in a slower passage of time compared to a region with weaker gravity.
At the center of the Earth, although there is no net gravitational force because the gravitational forces from all directions cancel out, there is still a gravitational field. This gravitational field induces a gravitational time dilation effect, causing time to run slightly slower at the center relative to regions further away from the Earth's center.
However, it's important to note that the gravitational time dilation at the center of the Earth is extremely small, and its effects would be negligible for practical purposes. The time dilation becomes more significant near massive objects like black holes or in intense gravitational fields, such as those near neutron stars or in the vicinity of massive celestial bodies.