What would happen if a character came in contact with or even became immersed in brine from an Illithid Elder Brain's brine pool?
Volo says little:
Volo's Guide to Monsters, the only authoritative 5e source where we might expect to see anything, only has
an elder brain sprawls within a vat of viscous brine, touching the thoughts of creatures near and far,
the creature dwells in a dimly glowing brine pool, filled with foul and brackish water infused with the elder brain's vital fluids and with psionic energy. (Both p.173, emphasis mine.)
So there's certainly daylight to steer yourself toward making up some sort of psionic effect that might occur when contacting elder brine. But there's nothing authoritative--in 5e lore or in the elder brain's stats, lair actions, &c--to say that elder brine itself has any effect.
According to R.A.Salvatore, there's no effect.
In Exile (published during 2e) we find a protagonist enslaved by mind flayers. Assigned to massage the illithid community's central brain, where
only a very few slaves were allowed within this special tower, captives with sensitive and delicate fingers that could massage the illithid god-thing and soothe it with tender brushes and warm fluids.... Drow elves always had proved skilled at this task, and this latest captive was one of the finest so far. (pp. 207, 219)
We see from Drizzt's time serving that the elder brine has no notable effects. Perhaps he's more-attuned to the elder brain's needs, but that seems to be shaken off as readily as the mind flayers' dominion.
As for immersion, I'll point you to PHB p. 183 for rules on drowning. =)
It seems reasonable to imagine a pool charged with psychic energy might stun those dropped into it, much like a mind blast. But that isn't the problem....
Everyone seems to be forgetting something critical about Elder Brain pools.
An Elder Brain pool is literally swarming with Illithid tadpoles (see the description in the MM or VG) – if you fall into the pool or are drawn into it by the Brain's tentacles, you will be exposed to thousands of tadpoles all over your body, and chances are high that you will either swallow one by accident or have one crawl into an ear... and that is the first step in ceremophosis into an Mind Flayer. Once a player is exposed to a tadpole, the character has less than an hour to live. Once the tadpole dissolves its way to the brain, removing it is nearly impossible. The character is doomed to a truly horrible death.