If I paint wood, for outdoor use, while it is still very wet, what problems can I expect in the long run (2-3 years)?
I think the issues that should concern you are more short term than long term, and that is failure of the paint and not a problem with the wood.
However, I think this might be a case where nobody can tell you for sure what's going to happen. Although obviously being cautious and waiting wouldn't be a bad idea, there's nothing to say for sure painting early the paint will fail. It's just more likely, not a sure thing.
Advice on this front, whether based on individual experience, book learning or manufacturer literature, will largely agree that the shouldn't paint onto a wet surface. But what does 'wet' actually mean? Wet and dry are in many ways relative terms, not absolutes. For example say you were working with wood that's at equilibrium with the local environment, of course that environment dictates how dry the wood actually is. So quite obviously in Georgia wood is significantly higher in moisture than in Arizona. The EMC in the first case could be well over double that in the second, yet in both places paint can perform as it's intended to.
I intend to paint every side, which will completely seal it with paint.
That's 'seal', not seal.
Only a coating that actually provides a good moisture barrier should be thought to seal a surface (and that is with a uniform, relatively thick, application).
"Latex" paint (terrible name — doesn't contain latex and never has!) is flexible and highly moisture-permeable, so actually it's an ideal paint to use if you want to paint wood that's not as dry as you'd wish it to be.
That's in respect to the wood and allowing it to continue to dry, but all waterbased paints rely on going on to relatively dry surfaces to bond well. This doesn't mean the wood has to be dry through-and-through, but the surface at least should be fairly dry or it will retard the drying of the paint and could significantly compromise bond strength and/or film strength (somewhat similar to painting at temperatures below the low end of the recommended range).
Slightly off-topic so I'll tack this on last:
The paint is an exterior latex paint with primer (Behr Exterior Premium Plus Ultra).
I would be wary of a product that is said to combine the qualities of primer and paint in one. All of them are compromises of one sort or another and I can assure you that in nearly every case that they cannot match the performance of a true primer followed by paint.