According to our current understanding of physics, as described by Albert Einstein's theory of special relativity, time dilation occurs when an object moves at speeds approaching the speed of light. However, it's important to note that special relativity prohibits objects with mass from traveling at or faster than the speed of light in a vacuum. Therefore, the concept of a warp drive that allows faster-than-light travel is purely speculative and not supported by our current scientific knowledge.
However, in some speculative theories, such as the Alcubierre drive, which is often invoked in discussions about warp drives, the idea is to create a "warp bubble" that contracts spacetime in front of the spacecraft while expanding it behind. This would allow the spacecraft to effectively ride a wave of contracting and expanding spacetime, potentially enabling faster-than-light travel without violating the speed-of-light limit locally within the bubble.
If such a hypothetical warp drive were to exist, it could potentially involve extreme distortions of spacetime. In such a scenario, it's conceivable that time dilation effects could occur within the warped region. Time dilation is a consequence of the fact that spacetime is not fixed but rather influenced by the distribution of mass and energy.
However, it is essential to emphasize that the concept of a warp drive, including the associated physics and consequences, remains highly speculative and purely theoretical at this point. We do not have a complete understanding of the physics involved, and there are numerous technical challenges and paradoxes associated with faster-than-light travel that would need to be resolved.