As far as I know, no refrigeration is needed, but it couldn't hurt. I remember reading something years ago that said it was almost impossible to overdose on it, you could drink a gallon of the stuff and be fine.
A guy worked for my dad for decades who chugged the stuff like he was a heroin addict. All he probably needed to do was quit drinking, and he wouldn't have needed it. He would sit at his desk and every 15-20 minutes pick up the (big) bottle and take a swig of it. We used to laugh at him and one time his son, who worked there too, brought an empty bottle of it with him to lunch. He took a fake swig of it and one of the women we worked with said, "What? You take it too?". He laughed and said, "Hey, just going to be an alcoholic mess like my dad and wanted to get an early start on it!". It was funny, but sadly, it was true, he eventually became a drunk like dad. About 5 years before his dad died, he got off the booze, and the Pepto, and all he took was aspirin for his aches and pains. I liked him a lot better booze free than back in his drinking/pepto days. He was embarrassing on business trips. I have many memories of him puking up Pepto at a dinner.