Cooked hash browns will last in the refrigerator for up to 5 days before they start growing mold and go bad.
After 5 days in the refrigerator the cooked hash browns will likely be growing mold and should be thrown out if they do have mold on them or have grown other bacteria as well.
You can keep the cooked hash browns longer than 5 days if you freeze them instead and then the frozen cooked hash browns will last for 6 months to a year.
Although after a month the cooked hash browns will start to go stale and not be as fresh tasting but they will still be good to eat.
So if you don't plan to eat the cooked hash browns within 5 days of refrigerating them then you should really freeze them to keep them from growing mold and going bad.
I have kept frozen hash browns for 6 months before I ate them and they been okay although not as fresh.