I'm guessing no, but I wanted to ask. Also, do you have to pay any maintenance (i.e. do you have to pay the various commonality fees? Again, my guess is no.
The Maintainance depends on you because usually ull get leased planes back be4 the D check so u have to make the check or noone will rent it ... so usually yes u have to pay the maintainance if u wanna still rent it out after its 8 yrs old ...
Short answer is no - for as long as it's leased out you don't pay any running costs.
As Athriaxo points out, smart lessees will often return a plane just before a D (or sometimes C) check as it's cheaper to terminate the lease a bit early than pay C/D costs.