Imagine a spaceship capable of achieving faster than light travel uses several transporters to allow crew members to move around its interior at the constant speed of light. What happens to the crew using the transporter when the spaceship goes to warp 2? BTW warp 2 means twice the speed of light in this context. Is there a clever way to work around this issue or I need to update the operation manual correcting warp 2 to move only at 20% speed of light?
The real answer is "whatever you want." We have no known physics for what happens past the speed of light, so you are writing your own rules.
However, if we are to use our own intuitive laws to try to write what happens faster than light, most of our laws of physics state that if every point that you care about is moving at the same pace, the laws of physics work as though they were all standing still. Thus you should expect the most natural behavior of light possible. No issues at all here.
If you wanted to work relativity in, light travels the same speed for everyone, so everyone on the ship should see light moving at the speed of light in their reference frame. (This unintuitive behavior is where effects like time dilation and space compression form).
The only thing which is "different" is that the whole contraption is traveling at fictional speeds. You write the rules here. However, if you want to draw from real physics, you can use supersonic aircraft as an excellent example.
In regions where everything is traveling at subsonic speeds, air behaves like a fluid obeying the rules of wave mechanics. Information of an event ripples outwards on these waves in a very intuitive way. Go supersonic, and the story gets more interesting. The air near the aircraft travels at subsonic speeds with respect to the aircraft. All the normal wave mechanics behaviors work close to the craft. The air far from the aircraft travels at subsonic speeds with respect to the ground (as you would expect). In between, there's a region a few nanometers wide where the fundamental assumptions of wave mechanics simply fall flat. The air behaves differently in this region. The result is a slightly more complicated version of a high pressure* front which handles all of the odd discontinuities between these two regions. This thin layer is what forms a sonic boom. This thin layer is also responsible for consuming an astonishingly large number of CPU cycles in simulation, so if predictability is something your characters need in your FTL scenarios, that could play a major part.
You could do similar. Have light travel "at the speed of light" everywhere except in a thin shock cone around the craft, where light does something different. It does whatever is required to stitch together the discontinuity that is caused by traveling faster than light.
After all, light isn't a particle. It isn't a wave. It's just light, and if you blink, you'll miss it!
* In this boundary region, "pressure" gets to be tricky. We typically assume that pressure is the same in all directions. This is critical for deriving the wave behavior of sound waves. In the bow shock of a supersonic aircraft, that breaks down. "Pressure," if you call it that, becomes a directional thing. Needless to say, that complicates things rather quickly.
Warp 2, eh? Well, you're clearly using a warp drive, like some actually functional and practical descendant of Alcubierre's ideas. Such systems don't involve the ship travelling faster than light at all, but only the warp in space around it. That's why at sublight speeds the occupants of a ship using a reactionless space-warping drive don't experience any relativistic effects like time dilation.
It also means in this context that beaming at lightspeed from bow to stern of your ship is exactly the same with the warp drive on as it is off.
If whatever FTL trick you're using affected the inside of the ship, it wouldn't be very healthy for the crew. It's not just teleportation. Imagine you are walking forward or backward in the ship. What would happen to you if you could experience the warp speed you're travelling at?
Obviously, any transport method used necessarily mustn't be felt inside the ship. Your transporters, however they are working, will do so exactly as on Earth (which isn't standing still either, btw.).
Try not to think about it too hard. It's not possible in this universe to go faster than the speed of light, it is irrational, so justifying how transporters will work in a rational way is also not possible.
The answers you have here that say that it would make no difference are all completely wrong in RL btw. They are based on notions that only work at slow speeds and with objects that have mass. In a universe governed by relativity, one cannot add one's speed to the speed of light -- that is, if you are travelling at 1000kmh, then the light bouncing off you isn't doing c+1000kmh, it is still only moving at c. This is partially because of the nature of photos (they don't actually 'bounce' off things but are rather absorbed and reemitted) and a misunderstanding of what the speed of light represents (clue: it's named badly, it isn't a speed limit, it's actually about the fundamental nature of reality).
If I were writing a sci-fi book (and I am) I wouldn't have anything travelling faster than light, just like Alistair Renyolds' Revelation Space. Relativistic speeds are sufficient for most purposes. You can circumnavigate the whole universe in a little over 50 subjective years if you want, without breaking the 'speed limit'.
I don't have a problem with teleportation though. That's just an engineering problem, though a moral/philosophical one too, obvs. -- you die every time you step into a transporter, like in Dan Simmons' Hyperion Cantos.
The best way to write sci-fi is to stick to what's scientific. Otherwise it is just another boring fantasy novel. (not that you said that you're writing a sci-fi book, it's just that whenever I read about teleporters and faster-than-light travel I assume the author is aiming for sci-fi, despite being so far outside the realms of science that you may as well put elves and witches in it).
Another approach would be to look at the current theories of "realistic" teleportation. They basically amount to the belief that it isn't viable to actually transmit matter like a Star Trek teleport, instead teleportation would work by mapping a body, killing you and then effectively printing a new copy wherever you want to be. At which point the speed you're moving doesn't matter.