Will there ever be a game where PS4 players can play with or against XOne players?

Also it would be kind of cool if there was an MMO-esque game where instead of having a faction, you play on PS4 or XBox One side :D

Probably, yeah, but most games will simply not be cross play.

No not likely.

Doubtful. MS has made it clear several times that they want the Xbox Live ecosystem to be it's own thing. Unless something has changed, they don't even want to be cross platform with PC.

Doubt it, but if it happens it'll be with PS and Nintendo.

Highly doubt it.

360 xbone
or
ps3 ps4

MAYBE.

playstation xbox, never.
you cant combine two account systems into one and expect it to work, thus a game would need its own accounts for the cross platform play, which people hate doing.

besides that, why in the world would xbox or playstation want to aid the apposing company in such a way?
Never gonna happen

There already kind of is. For Portal 2, PS3 players can play with PC players, I believe.

Also, isn't there that chess game which has some cross-platform functionality?

(Between Xbox and PlayStation, very unlikely)

I think Final Fantasy has PC and PS4 players together. As does War Thunder IIRC

Never. If there were to do that they might as join force and create one system to play all their games under.