According to the current understanding of cosmology, the expansion of the universe does not involve objects moving away from each other at speeds greater than the speed of light. It is important to note that the expansion of the universe is not a movement of matter through space but rather a stretching of space itself.
The concept of time dilation, as described by the theory of relativity, applies to objects that are moving relative to each other at significant fractions of the speed of light. However, the expansion of the universe is a global phenomenon that affects all points in space uniformly. It is not a motion of matter within the universe but rather a fundamental property of space itself.
As a result, the expansion of the universe does not lead to time dilation effects between objects due to their relative motion away from each other. Instead, it manifests as an overall stretching of space and an increase in the distances between galaxies. In this context, the expansion of the universe is not subject to the same relativistic effects as objects moving within space.
It's worth noting that while objects within the universe can move away from each other due to other physical processes (such as the motion of galaxies within galaxy clusters), these motions do not result in velocities exceeding the speed of light. The theory of relativity still applies to these relative motions, and time dilation effects would occur if the velocities involved were significant fractions of the speed of light.