The dehumidifier will run, but will not collect any moisture from the air. The screen has been cleaned, the tub is cleaned, but no water will collect.

A dehumidifier is basically the same thing as an air conditioner but both coils (condenser and evaporator) are in the same unit meaning a dehumidifier exhaust warm air in the same room it is in. running a dehumidifier would make an air conditioner run more because of the warm exhaust, when you run an air conditioner it acts like a dehumidifier and dries the air. As far as saving energy it all depends on the size of the AC and dehumidifier and the size of the room. If you want warm dry air, use a dehumidifier; if you want cool dry air run the AC but not both.
How do all these parts fit together to pull moisture from the air? It's fairly simple, but very effective:

1. A fan collects air from the surrounding area and pulls it into the dehumidifier.
2. As the air passes through, it comes into contact with the dehumidifier's cooled coils. These coils use condensation to pull moisture from the air. The collected moisture remains on the coils and drips into the dehumidifier's reservoir.
3. The dehumidifier reheats the air and exhausts it back into the room.

A dehumidifier usually has a removable plastic bucket for are reservoir; most buckets also have a place where you can hook up a hose so the collected water can drain straight into a floor drain or pump. This frees you from having to remember to dump out the water. But don't worry too much about the reservoir overflowing -- most dehumidifiers also have an automatic shut-off. If you're using a dehumidifier in extremely moist conditions,however, or if you need to keep your dehumidifier on all the time, you should look into a unit with a built-in condensate pump, which regularly pumps water out of the unit's reservoir rather than simply relying on gravity to empty it as a hose does.