Polyjuice Potion seems to last at least 8-12 hours (I think this is obvious, but I'm just going by canon). In Chamber of Secrets in chapter 12, The Polyjuice Potion, Hermione is up "very early" on Christmas morning, adding lacewing flies to the Polyjuice. She suggests that the trio take the Polyjuice that night, which, depending how early she was up in the morning, gives the 8-12 hour window, give or take. And I think the events of Goblet of Fire probably demonstrate that Polyjuice has a decent shelf life. It would be hard to consume an entire cauldron of Polyjuice super quickly (although Polyjuice has to be drunk on the hour for ongoing effects.). In chapter 21, The Unknowable Room, in Half-Blood Prince, in March, Harry suspects Draco Malfoy of having stolen Polyjuice Potion from Slughorn on the first day of classes the past September. We subsequently learn that Draco has been giving Crabbe and Goyle Polyjuice Potion so they can act as Malfoy's lookouts while he works on the cabinet in the Room of Requirement. So altogether that gives a window of 8-12 hours to 5-6 months minimum shelf life.
Veritaserum is another potion that could have a solid shelf life. In Goblet of Fire, chapter 27, Padfoot Returns, when Snape threatens Harry with Veritaserum, he mentions the use of Veritaserum is highly regulated by the Ministry of Magic. This could be interpreted as the Ministry controlling most supplies of Veritaserum and doling them out when appropriate. Moreover, the events of Order of the Phoenix suggest Delores Umbridge used Veritaserum on students throughout the course of an entire school year, or close to it. As Veritaserum is not a fast-brewing potion, Snape had to have stores of it. Snape says to Umbridge that she used up all his stores of Veritaserum. So I estimate Veritaserum has a shelf life of at least ten months (the length of a school year in Harry Potter)
Hermione notes in chapter 15, The Unbreakable Vow, in Half-Blood Prince, that there are at least one dozen different love potions. As they all are being sent from Weasleys' Wizarding Wheezes, I imagine their shelf life is good and, as TGnat and NominSim both first mentioned, strengthen as they age. I believe Amortentia has a pretty strong shelf life as well, as the Love Room in the Department of Mysteries has a large fountain inside that flows with Amortentia [POTTERCAST TRANSCRIPT] Now, volume wise, that would be a lot of potion. It would seem impractical to have a fountain flowing with a potion the Unspeakables would have to brew super frequently.
It seems clear that Felix Felicis has a shelf life of at least 9-10 months, as Harry wins this potion on the first day of classes in September and takes it the following June.
Has anyone mentioned the Horcrux potion in the sea cave? Clearly this potion has an incredibly long shelf life. It's unclear whether the Horcrux potion is a regular potion or if it would be considered of the Dark Arts. In What Order Were the Horcruxes Made In, I estimated that Voldemort created the Slytherin locket Horcrux between 1945 and 1946. Deathly Hallows shows Voldemort putting the locket in the potion basin in 1978, when Regulus Black was seventeen. Regulus drank the potion in 1978. Somehow the basin was refilled with potion, which sat untouched for nineteen years until Dumbledore drank it in June of 1997. So we know the Horcrux potion has a shelf life of at least nineteen years.
Wolfsbane seems to have a shelf life of at least 1-7 days. In Prisoner of Azkaban, Lupin explains that he drinks Wolfsbane in the week prior to the full moon. In Half-Blood Prince, he confirms that Snape did indeed brew the Wolfsbane potion for him every month. So I suppose it couldn't last longer than however many days are in a complete moon cycle, but it must last long enough for the werewolf to drink it during any of the seven days preceding the full moon.
‘The Potion that Professor Snape has been making for me is a very recent discovery. It makes me safe, you see. As long as I take it in the week preceding the full moon, I keep my mind when I transform ... I am able to curl up in my office, a harmless wolf, and wait for the
moon to wane again.
Prisoner of Azkaban - page 258 - Bloomsbury - chapter 18, Moony, Wormtail, Padfoot and Prongs
‘But I do not forget that during the year I taught at Hogwarts, Severus made the Wolfsbane Potion for me every month, made it perfectly, so that I did not have to suffer as I usually do at the full moon.’
Half-Blood Prince - page 312 - Bloomsbury - chapter 16, A Very Frosty Christmas