Do you mean closed in the theoretical sense, or just closed in the common sense. Because even though you can't see out of a closed elevator when it is moving, any signal that gets in can be used to measure motion, provided you know that the signal is stationary (or, if not, you know how it is moving). A sensitive enough meter to measure the magnetic field (for instance), could tell you how the elevator is moving with respect to the Earth.
Another technique, if you can take a reading before the elevator is sealed, is to use inertial measurements to keep track of your movements. This is how rockets' and weapons' guidance systems work.
But you have a problem of definition. You ask how you will know the lift is moving... but you never say relative to what. "Moving" means nothing without a frame of reference.
If everything inside the lift can't interact with anything from outside the lift, then you'd have no way of detecting whether the outside is moving at 10000m/s or 0m/s, so you can't determine the "motion status."
If you can somehow interact with stuff outside the lift (e.g. a magnetic field as Indi said) then you may be able to determine how fast the lift is moving.
Implicitly I've assumed moving means moving RELATIVE to stuff outside the lift. This is the matter of definition Indi mentioned. For example, if I'm walking forward at 5 m/s, you could say I'm NOT moving but that the Earth is moving backward at 5 m/s.
You just listen for the mechanical noise of the pulleys and cables moving it of course! ^.^
In addition to magnetic fields, you could also measure gravitational fields, with sensitive enough measurements.
If there is no gravitational field present, you've at least solved part of the problem: you still don't know if it is at rest or moving, but without gravity, asking if it is going up or down is meaningless.
Please Share your ideas 
