No, the Xbox 360 Elite does not need an intercooler.
Intercoolers are for turbocharged/supercharged car engines—they cool compressed intake air. The Xbox 360 is a console, not an engine.
The Elite has an internal cooling system (heatsinks, fans, heat pipes) designed for its thermal load. It's sufficient for normal use.
If you experience overheating (Red Ring of Death risk), consider:
No intercooler required or applicable.