Gravitational time dilation is incredibly weak on Earth, since Earth is not very massive on cosmological scales.

Okay well let’s just stipulate that the object is travelling close to enough to the speed of light for there to be time dilation of some sort.

There is always time dilation between any two frames of references moving at nonzero speed with respect to each other. It’s generally negligible for everyday velocities, but it’s still there. You can find the degree of time dilation (and length contraction for that matter) in special relativity (i.e. ignoring gravity) by computing the gamma/Lorentz factor. For example, for 90% of the speed of light, the Lorentz factor is about 2.29.

Or maybe the object is stationary but near a black hole or something so there is time dilation from the gravity

In that case, it depends on how strong the gravitational effect is. The mathematics is a bit more complicated though. I would recommend to stick to special relativity if you’re learning about relativity as an interested layman.

This hypothetical is of the type “immovable object versus unstoppable force.” The question becomes: which of the axioms of relativity do you want to discard? Yet, once you do, you are leaving the realm of physics and entering the realm of sci-fi, in which anything may be possible.

If you want to maintain any link to our current understanding of physics, there are no hypotheticals, no ifs or buts. It’s simply not possible to have a set of laws of physics consistent with relativity as we understand it, wherein massive objects can travel at the speed of light in vacuum.

Physicist here. Many common misconceptions in the comments.

1. No, someone traveling at light speed won’t arrive “instantly” or anything of the sort. It’s simply not possible for massive objects to travel at the speed of light in any valid (inertial) frame of reference. Any system that does travel at the speed of light (e.g. a photon) does not have a frame of reference in which it is at rest - instead, it moves at the speed of light in all frames of reference.

If the other person travels at some speed (just) below the speed of light, the signal they send will be Doppler shifted/time dilated according to their relative velocity.

2. No, quantum entanglement cannot and never has been used to communicate faster than light. See: no-communication theorem.

Most likely a contemporary film that neither features history, nor science, nor technology. Hollywood rom-coms are good candidates since they tend to be quite shallow.

Because empirical evidence is lacking.

Not at all. Yes, there are a few such as Penrose and Josephson who have dabbled in this nonsense, but very few in the community take it seriously.