Does leaving a flat for months damage the wheel?

Leaving a flat for months won't damage the wheel assuming the flat is the only factor in the equation.

If the tire going flat was part of a chain of events there might be damage.

If the tire was sitting in a half inch of water, the tire went flat and the steel wheel was now sitting in water the rim might rust. Even in this example I doubt the rust would be bad enough that the wheel would be unusable.

The tire takes damage from not being inflated when a bike is standing on it. It deforms, and its structure may be weakened. It might be possible to remedy a deformed tire, but if it develops cracks on its flanks after being inflated, the tire needs to go into the trash. You say that you want to replace your tire anyways, so this is not your problem.

The rims are usually aluminum today, and spokes are typically steel, both of which don't care whether they are rotating or not.

What you need to worry about is rust and lubrication of all moving parts, though. Your bike has not been moved for six months, it might need a bit of maintenance before delivering 100% performance. This includes the bearings within your hubs, though they should be fine. Just check that your wheels keep rotating freely for a while when you spin them up a bit. The longer they keep turning, the better. The most important part to check is the chain, it may have become a single piece of rust in the meantime. Usually, a few drops of oil and a bit of spinning should get the chain flexible again, but a high humidity environment may have killed it. Bottom line: The exact amount of maintenance needed can only really be judged by looking at your bike itself.