You can put whatever you want down. The less you put down, the higher your monthly payment will be.
There will be some fees, so you won't be able to do it for nothing out-of-pocket. But unless having a lower monthly is very important to you, I wouldn't do anything more than $1000 down toward the lease. Some would even consider that excessive. Save/invest the cash you would put down, and it should more than make up for the higher monthly.