Would this use a large amount of mobile data?
Would having this running in the background use as much battery power as say Google maps (without ever having to display a map)?
Note:It's okay if the app is measuring in intervals (every 1/8 mile) if that would help reduce the data or battery useage...
Assuming you are looking for software, not in-car telematics, and you are willing to integrate rather than build it yourself, you could try using ZenDrive. That would give you a lot more than mileage too (eg, driving behavior).
This should neither use a large amount of mobile data nor use as much battery as Gmaps. Can't speak for iOS but this is what the Android documentation below has to say on location strategies and what it boils down to is that you need to determine how much accuracy your app needs, pick your strategy and maybe implement it as a background service that talks to your app. Someone who coded for Android before should have a prototype up and running within a few days. I assume however that this involves extensive testing on various devices and OS versions (the tricky part of mobile development).
Yep, you would be polling intervals/speed and saving those locally during a drive session then you would post those numbers back to the server for calculation. No need to show a map.