|The typical movie jump to hyperspace|
The reality is not quite so visually exciting, but it is still impressive and decidedly confusing until you think through what is happening.
As a ship accelerates towards the speed of light, two things should happen. One is that there will shifting of colours. The colours of the stars behind the ship will be red-shifted, moving down the spectrum and those in front will be blue-shifted, moving up. This means that some stars will disappear as their colour goes out of the visible range, while others will pop into visibility for the first time.
The second thing is that the stars will move towards the front of the ship, bunching up in the direction of flight (though still as points, not the traditional movie streak). As the ship gets close to light speed, even stars that were almost directly behind it will appear in front. This is the least intuitive aspect, because it seems more natural that the starlight will be 'left behind' than brought to the front. What you have to bear in mind is that while the light was travelling from a particular point, you will have moved with respect to it. Add in the relativistic complication that light continues to travel towards you at the same speed whatever speed you are moving at and you end up with the bunching effect - it just remains very difficult to envisage. This website may help.
Things get more messy when we enter warp drive, whatever than means, as what you see is likely to depend on the mechanism of the warp drive itself, and since they are almost all imaginary, it's really rather up in the air. This website suggests that it would basically be more of the same - that in the warp bubble envisaged for a real warp drive you would see the same effects as when nearing the speed of light (but even more so). However the environment of a warp drive is quite different from a ship travelling at near the speed of light. Technically in a warp drive, the ship isn't moving at all, it's space that is moving around it. And once you get into the more dramatic science fiction concepts like hyperspace it isn't at all clear what the implications would be - quite possibly, there would be no stars visible at all.
There is always a tension in science fiction movies between getting a good representation of the best science tells us and making the film work as a piece of storytelling. Often the reality is ignored due to ignorance or a misapprehension of the audience response. As I mentioned recently, I think the silence of space worked superbly in 2001, and yet practically every movie since has given us sound in a vacuum. In the end, the Star Trek/Star Wars whoosh of stars as warp/hyperdrive is engaged and similar effects probably don't do any harm (though I here and now put a scientific curse on any movie or TV show that portrays a star field that moves visibly past as the ship flies), if only because the technology is fictional. But it doesn't do any harm to ponder it.
Image credit: University of Leicester via space.com