'In the lab' could mean that the lab has more than one room, and the locks are damaged on doors leading from one room to another, inside the lab. Or it could mean that the locks are damaged on cupboard or closet doors inside the lab (whether the lab itself has more than one room or not). 'At the lab' could mean the same as above (either option), or it could mean that the locks are damaged on doors leading into the lab from outside of it. Either way, 'the lab' is correct, not just 'lab.'

Would it be correct to say: Door locks are damaged AT power system lab?

I suppose it depends on where the doors are.

Which one do you think is correct? You have a 50-50 chance, so just guess. Do you have other examples of in or at? School, home, house, store, etc. Also, in both cases, you need the word 'the', in the lab or at the lab.