Can an unpowered DC motor hold up a weight?

I am planning to use Arduino to drive a DC motor with a pulley that will raise the pop hole door of a chicken coop. This will be like a crane lifting a load.
My question is when you remove the power from the motor will the weight, in this case the door, drop back down to the closed position or will it stay were it is until the door is lowered by driving the motor in reverse?
I am thinking of using a small (few hundred mA) motor with a 50:1 or 100:1 reduction gearing and a very small pulley say 4mm radius and a door weight of 1 or 2 lb.
Any ideas how I work out if the weight will turn the gears and motor please?
I have the idea in my head that the gearbox will not turn, I recall being told that the gearbox would be damaged before the force to turn the output would turn the unpowered motor but it would be nice to know before I choose the motor.
Thanks.

With gears it is very likely that the motor will be backdriven when it is unpowered. If you're looking for a mechanism with unpowered holding capability, look into worm screw transmissions (example, power windows in a car).

Or make a mechanical design such that when the door is open there is little or no rotational torque being applied to the gear shaft.

I think what zoomkat is suggesting may be to put a counterweight on the mechanism so the door is more-or-less balanced. Then the friction in the gear train is likely to be enough hold the door open.

And, as an added bonus, it’ll take less energy from the motor to raise it.

Think “lift bridge”.

Brilliant. Thanks.

I will look into worm drives but I think a counterweight will be easiest for me; the walls of the chicken coop are double skinned and the weight can live in the cavity.

if you get something compact with a high gear ratio it'll likely be a worm drive just for space reasons.

I will look into worm drives but I think a counterweight will be easiest for me; the walls of the chicken coop are double skinned and the weight can live in the cavity.

This is the way large windows were counter weighted in the old days when windows actually opened and closed (aka, before airconditioning).