If indeed we are able to make something move close to, at or faster than the speed of light, many strange are theorised to happen. One DifferentTurret pointed out - mass is gained at an exponential rate. Another is time changes significantly - this falls in the line of special relativity, which is too long and complicated to properly explain. But there are theories that if you move past the speed of light, you can actually reverse time, which does makes sense if you understand special relativity (of course it's just a hypothesis that you could reverse time so while it makes sense doesn't mean it can actually be true!).
However, it would be very interesting to see if they could make a photon move faster than the speed of light, so many theories about time travel could be tested, huge leaps in science could be made if it is possible. Of course if it is possible, it might not be able to be perceived, how much of mind-fuck is that?
Wait, I've heard that we would go FORWARD in time if exceeding the speed of light?
Anyways, there are alternate ways to covering distances faster than light can. A popular theory of course is that black holes are wormholes. Here's how it would work without breaking physics: Imagine a paper, this paper representing the fabric of space-time, with two points on it labelled A and B, much like this:
Remember, the graph represents space-time. This will be important
No matter what, you must cross from point A to B by crossing the distance from A to B, right? Well, there's a way to get to B faster than light could from A without cheating. Einstein established in the early 20th century that mass creates a bend in space time in the form of gravity. Examine the GIF below.
Now, that is a star. a black hole's looks like this:
Now, go back to the paper. Imagine that someone took the paper and folded it such that A and B are now right next to each other. Using a very powerful gravitational object with a lot of mass, such as a supermassive black hole, we could do that to the fabric of space time. This is how it would be illustrated:
According to this theory, the speed of light may be the limit, but there's a way around it. Of course, a white hole has yet to be discovered, which is why this is a theory.