No, the only thing to do with the older games are a few characters in the game.
Dante, Trish, Lady and Sparda, the first three all appear with you controlling Dante for half the game, while Sparda is mentioned and worshiped as a God in the games setting.
Other than that you don't need to have played the other games, but their awesome, so you should get them anyway :P
Credit to .Impact for the sig & avy