After reading this question (How did the Death Star move?) it sounds like the Death Star moves by firing ion drives around its equator. The answer also states that it can move faster than the speed of light.
There is a non-canon example in the X-Wing game's companion story book/strategy guide. It included a novella called The Farlander Papers where an Imperial frigate decelerating from hyperspace hits an X-Wing fighter. The X-Wing was obliterated and the frigate was unharmed. It was explained that the frigate's shields protected it from such collisions.
Did it happen? No.
In current (Disney) canon, there are only two moon-sized or larger hyperspace-capable ships: the Death Star (I and II) and Starkiller Base. Neither of them has collided with anything in hyperspace in any of their canon appearances. There may be Legends examples, but I can’t find any.
Is it possible? Yes.
It’s important to distinguish between two related but distinct situations here, either of which could be considered as “colliding at lightspeed.”
An object could emerge from hyperspace close to another body and collide with it. It would be traveling very fast, but not at lightspeed; faster-than-light travel is only travel is only possible in hyperspace. This has happened, but never to a ship the size of the Death Star.
If destruction of the base was the goal, why hadn’t whoever was behind the attack used the ship as a bomb by reverting from hyperspace in closer proximity to the moon? Planetary bodies larger than Sentinel had been shaken to their core by such events.
If this happened to the Death Star, it might be destroyed, or might not be. Its shields could protect it from collision, depending on the mass of the colliding body, and possibly whether the colliding body had shields.
An object could intersect another object’s gravity field in hyperspace. Such fields are referred to as gravity wells, though there are also free-standing gravity wells. This, again, has happened, but not to a Death-star-sized ship.
Another, the Arbitrator, made a bad hyperspace calculation to escape pursuing NR ships. It evaporated when it was sucked into a gravity well.
Aftermath: Life Debt
If this happened to the Death Star, it might well be destroyed.
It's not canon, but in Star Wars Infinities: A New Hope, The Death Star is piloted into Coruscant.
The station is obliterated, dealing massive damage to the planet in the process, though it is unclear if Coruscant is entirely destroyed.
This is the only instance of a Death Star-sized ship crashing into something comparatively large in an officially licensed work that I am aware of.
Same thing that happens when anything gets hit, the large thing takes damage, the small thing gets pulverized. Granted it would be unlikely that the Death Star would be hit with debris that could damage it seriously. At sub-light speeds the Death Star has limited shields and arrays of turbolasers intercept or deflect debris that pose a threat. As for FTL, Star Wars ships use Hyperdrives that take a ship out of normal space and into hyperspace. It is how they avoid dealing with impacts at luminal velocities.