Neostigmine is a real-life drug known as a cholinesterase inhibitor. It has the ability to counteract tubocurarine, which is a potentially fatal muscle relaxant capable of shutting down the respiratory system of the recipient's body. In this way, it is an antidote. It is a type of carbamate compound which acts as a reversible acetylcholinesterase inhibitor, exciting the parasympathetic nerve.
It is located in the Treatment Center's safe in a vial delivered with an injection gun.
Virtue's Last Reward
Neostigmine's only role in the game is the key for the plot lock leading to K's ending. The drug itself is never used for any significant purpose. Escaping through the Number Nine Door with insufficient BP is one of the uses suggested by those who discover the neostigmine. However, by the time neostigmine is brought up, nobody remains alive long enough to make use of it (Clover END) or it is used for a different purpose (K END).
Neostigmine might have been a plausible option to save Sigma's life (and his arm) in Quark's ending since the treatment center was one of the explored rooms. However, nobody from the team who explored it mentioned finding the injection gun and most likely had no idea what it was anyway.
Tenmyouji may have found and used neostigmine in Luna Game Over 1. He opened the Number Nine Door using an unconscious Quark's bracelet to allow Quark to escape. He decided to accompany Quark through the door despite having only 4 BP. He would receive a penalty and most likely die before Quark woke up unless he injected himself with neostigmine. The possibility is small however, since the way Tenmyouji spoke in front of the door made it seem like he was okay with dying as long as Quark was safe from a potential murderer.
Sigma and Clover find an injection gun in the treatment center's safe. The gun is loaded with a vial containing a red liquid labeled "neostigmine". Clover states that although the name on the vial looks familiar, she can't remember exactly what significance the drug has. She later realizes neostigmine's purpose after discussing her backstory at length with Sigma in the Pressure Exchange Chamber. Her memory is triggered when she states that Free the Soul's goal is to "purify the unclean", which leads her to focus on the last word, "unclean", and finally the last syllable, "ean", which is also the last syllable of neostigmine. She then tells Sigma that it can counteract tubocurarine. She learned this information from her brother. She goes to tell Alice, but she succumbs to Radical-6 along with everyone else and kills herself in the Infirmary.
Phi and K cannot enter the white Chromatic Doors due to Dio's bracelet being broken. As the doors open, Sigma remembers that neostigmine can be used to counteract tubocurarine and that Dio took an injection gun from the treatment center's safe. He runs to Dio's treatment pod to search for the gun. After locating it, he takes it and rushes back to Warehouse B. He arrives just in time to watch the Chromatic Doors close. He, Phi, and K are penalized and injected with soporil. Since there is only one dose of neostigmine, Sigma must decide which person to save. He chooses to inject Phi since K's armor cannot be penetrated by the needle of the injection gun. Sigma and K eventually succumb to the tubocurarine and die.