Running lean simply means that the air/fuel mixture is not optimized and there is more air than there should be.
This means that the combustion in the cylinder is going to be hotter, hence, the motor will run hotter and be more susceptible to overheating.
Also, the air/fuel mixture is more susceptible to pre-ignition.
This means that the cylinder will fire before it's supposed to, causing a "pinging" or "rattling" noise when the engine is under load.
This pre-ignition, or pre-detonation, is what is most harmful to a motor. This is because the piston is not in the perfect position for the combustion.
The consequences of long-term and/or severe "pinging" are cracked pistons, wrist pin damage, and even holes in pistons.
Installing a CAI that leans out your motor is not in itself going to destroy it.
It's not great for it, but if you don't hear any pinging under load, then it's not bad enough to worry about.