Ok, so first lets disregard the fact of whether you or a phone could survive travelling that fast.

In theory it is possible to make the call on mobiles, which transmit microwaves and radiowaves, both of which are on the EM spectrum and therefore travel at the speed of light. As long as you were travelling at less than the speed of light, you should be able to make the call.

However you would then face the problem that sound waves travel considerably slower than the speed of light (around 900 metres per second compared with 300000000 metres per second). This would cause a problem as you would be running far too fast for your sound waves to reach the mobile.

I assume this is by mobile phone.  The radio waves being stretched relativistically, your phone would need to be retuned to a very low frequency.  The message would be similarly stretched out, so communication would be very slow.

We'll have to talk hypothetically here, since such speeds are impossible without particle accelerators. Firstly, relativity states that light ALWAYS recedes from you at 299 792 458 m/s. So in short, if you were travelling at the speed of a particle in the LHC, which is roughly 299 795 455 m/s, then fired a laser directly ahead, it would still appear to recede at that speed. Same goes for if you fire it backwards.

Therefore electricity and any EM waves will also abide by these laws. But, hypothetically, let's say that it travels as a constant 299 792 458 m/s, i.e. at the LHC speed light would appear to travel at 3 m/s.Aside from the problems of time-warping caused by speed, a phone call to a ship travelling close to light speed, which for convenience I will call c, could last a long time. The voice recieved on the ship would be really fast, since while 6 million years may pass on Earth, a day would pass on a ship moving at c.

Again, hypothetically, let's say time warp is not a reality. Then you have the problem of the raqdio waves moving towards you at a relative speed of 3 m/s. So again, this conversation might as well be taking place between Earth and the Andromeda Spiral.

Then there's the sound waves themselves. While light travels at c, sound only travels at 343 m/s - still fairly speedy, but sluggish in terms of c. So by the time the sound reaches the point where your ears would be, you are several thousand kilometres in that direction.

Overall, such a conversation would be impossible.