Very unlikely to be a problem - people use iPads with 3G and GPS receivers all the time. Don't leave the phone near the compass...
If you leave the mobile signal on - it will use extra battery due to it trying harder to find a signal. I've left mine in my pocket before with no issues.
There are many apps that can do this, on pretty much any phone which has a GPS receiver (I've an old Nokia and use sports tracker). If you can keep a signal in the car then you should be able to in a GA plane. Just set it up before you start, then forget about it until after you've parked the plane.