The novel I, Jedi touches on lightsabers, their construction, and their power. They have power cells that discharge very slowly (like a laptop that's turned off) and this power drain isn't significantly affected by simply being active - they are incredibly efficient.

The power is drained faster when something hits the blade, be it a solid object, a blaster bolt, or another lightsaber.

In The Krytos Trap, a novel from the X-Wing series, Corran Horn (pre-I, Jedi days) discovers a wing of a Jedi museum on Coruscant and finds several lightsabers in it. One of them, which he 'borrows' briefly, still works and helps him immensely in the culmination of the book's plot. Later, Luke talks with Corran about the museum and the lightsabers, and comments that the lightsaber Corran found was one of three in the entire exhibit that functioned without recharging.

This demonstrates that lightsabers DO have power cells that can run dry, even when deactivated, AND that this can take decades - the lightsaber in question hadn't been recharged since before the Clone Wars, and the novel was set post-ROTJ.