Gas and Oil Home Heating Furnaces - Heat Pump and dripping water

This is my first house with a single AC/heat pump unit for heating and cooling. I understand in AC mode that condensed water drips out of the unit, but is this supposed to happen in the heating mode as well? Thanks in advance.

Drips out of where? There should not be any "condensate" driping off the inside coil (Do you have a split unit: condenser outside and blower in/ or do you have single package unit, outside?)
If single package unit outside you may be seeing the water that defrosts off the condenser (outside) coil. You should not see any dripping inside .

It's a 2 part unit, and the terms I've been using are the compressor for the outside unit, and air handler for the inside unit. The water is leaking out of the outside unit, onto a concrete slab where it won't do any damage. Thanks.

Thats good, that is what it should be doing! If you go outside and look at the coil especially when its in the 25 to 40 degree range you will see frost building up on the outdoor coil if it is working good. It will then go into a defrost cycle and melt this off about every hour to 90 min. I hope they had some way of putting it up on a frame or heat pump legs, if not, sometimes the water will re freeze and a big ice cone forms inside the outdoor unit......all in all, the water is normal.

cool, thanks. I live in the phoenix, az area and the lows are in the 40s, so I'm guessing that I probably won't see any ice forming. In this kind of climate, do you think they'd disable the defrost cycle?

No you dont want to disable the defrost cycle. It will only go into defrost cycle if it needs to.;) ED

Your in a very dry climate, you won't see much, but it will still frost if the outdoor coil gets below 32degrees, right? Ice build up is not a concern to you though or even the heat pump legs...