I have a PS3 and want to play with and or against my buddy who has a PS4. Does this matter having different machines? It's all the same network, right?
According to this site: http://www.polygon.com/2013/10/30/5047272/sony-ps3-and-ps4-cross-play-is-technically-possible
It should technically be possible since both consoles connects to PSN. It's up to the developers to implement it.
Bottom line, I don't think this is gonna be possible for PS3 and PS4 COD3. Since users all over the internet are all saying that they both have different servers.
But that still depends on the game, it might be possible for other games.