According to everything we know today, including all experimental results and observations, nothing can be accelerated up to or beyond the speed of light. Light itself travels at that speed, and theoretically there's nothing against particles that are always faster (though we've never proven their existence). The problem is speeding something up to light speed and/or beyond. Relativity says that the faster something goes, the more massive it becomes (well known and measured, and even a necessary part of particle accelerator design). With more mass, you require more force to accelerate the object, and so ultimately at close to light speed, the object has nearly infinite mass and so requires infinite energy to be accelerated any more. Nearly every aspect of relativity has been experimentally proven so this result is pretty sound unless something new is discovered.

First, rest assured that the Sun won't explode. Low mass stars like the Sun toss off their outer layers with little violence, leaving behind a white dwarf.

The quickest that news of anything happening on the Sun comes to us is at the speed of light, which takes about 8 minutes 20 seconds to reach the Earth. If we lived around a more massive star that was going to explode, we may get a little advanced warning (perhaps a couple of hours) from tiny particles called neutrinos before the flash and blast.