Do I need to dry white oak before it can be used for an outdoor balance beam?
I think so yes. Green wood, and very much green oak, is used for building things but I don't think this project is a good fit for using it for multiple reasons, but perhaps most importantly because of various safety concerns.
One face of each beam is intended to be a walking surface, and there's no way to ensure that surface remains flat. And as the wood shrinks from green to air-dry over the next few years it will inevitably crack — and the number, severity and position of the cracks can't be predicted — so it's just a throw of the dice whether you'll get one forming on the top face of one or more of the members.
And be under no illusion of just how large the cracks can get. If any of the pieces have 'boxed heart' you can almost guarantee you'll get a radial crack (AKA heart check) and on 6x6s these can certainly be wide enough to trap a small fingertip or toe.
Here are some photos to show how bad the cracks in thick oak members can be, and this is even with management (in a mill's yard) or under cover/indoors, not uncovered in the open air and left to their own devices.
From the Comments:
If I really must use dried wood for this, then I'll have to find a different lumber source.
Obviously building this from dried oak would be a different story, but I doubt it's going to be economically viable in the current lumber market. Although there could be significant local variation the latest I heard (about two weeks ago) typical lumber prices have risen again to roughly double pre-Covid levels; and oak of these dimensions would already have been pricey before!
So for cost, availability and reliability reasons I'd suggest just going with with pressure-treated SYP as you were considering using for the posts. This sort of wood is what is most commonly employed in the construction of playground equipment anyway.