The Flask is an item initially taken by Mic Stone from the Orb agents in the Ferry Building. It says "REMEDY" and apparently it was supposed to cure him from his Ratman mutation and make him a human again. However it was a poison (perhaps as punishment for failing to returning the Viewer) and when he drank it in his hideout, he died and his body was rotten.
The Manhunter who followed his traces to his hideout in the Embarcadero Fountain, found the Ratman's body and took the Empty Flask from near his hand.
Later at the Doctor's House he filled it with some urine sample and became Full Flask.
Then at the Private Club he played a game with Ratmen who betted on body parts. The Manhunter offered them a Full Flask of urine and the three ratmen entered a fight. One of them drank it while the others were fighting, perhaps also hoped it was a remedy for their mutation.
Behind the scenes[]
If the Manhunter attempts to enter the Rat's Den sooner than he should, he stumbles down the stairs, drop the Flask, and it shatters. The game warns "You'll be dead if you go ahead" as the game can't be completed without it. The Flask must be used before proceeding there.