I think when there’s something there, like the locket, in the basin the potion can refill itself but if there’s nothing the potion loses that ability. Because Harry took the locket after Dumbledore drank the potion, there was nothing in the basin, so the potion didn’t refill itself.

But the locket Harry took was a fake one

Yes but it was still a locket. I think ‘cause of his arrogance, Voldy didn’t bother to set a charm to check if the locket was the real deal — he just set it in place to refill the potion if a locket shaped thing was there. That’s why it refilled when Reg kept the fake locket there.

In my mind it was less of a "drink until there's nothing left" situation but rather a "drink until you reach the locket" situation in which the drink of despair refills itself automatically. If you didn't grab the locket, the basin would refill itself

Maybe it’s that after awhile it just refills itself