I can confirm the following on four devices after testing things out this morning: (Updated all devices to 7.1 yesterday around 1pm)
iPad Air does not replicate this problem after updating to 7.1
iPad 3rd gen does not replicate this problem after updating to 7.1
iPhone 5 on Sprint does not replicate this problem after updating to 7.1
iPhone 5 on AT&T does have the issue after the 7.1 update restart occured.
In relation to the troubled phone, these are the details I have noted. The following GPS apps are causing the arrow to stay illuminated in the task bar:
Apple Store (This app is installed on all four devices that I tested)
International SOS Travel Security App
Foap
Siri
Google search app
TravAware Travel Security App
The following GPS apps however DO NOT cause the issue and are running location services just fine in the background:
AccuWeather
Apple weather app
Apple Camera app
CNN
Facebook
Find my friends
Groupon
ATT Wirless app
Yelp
Urban Spoon
Apple Reminders app
Open table
Navigon
Apple maps
Instagram
INRIX Traffic
Google maps
Fandango
Flight Aware
Apple compass
Camera+ app
Arround me
I have tried the following to resolve the issue but to no avail:
Restart phone
Reset phone
Deleted and reinstalled the troubled apps
Looks like this may be an issue that each app's dev must correct and then update. Or it's an issue with specific carrier phones but obviously we can't be sure yet without more input from others...
In my case, It's annoying but it does not seem to be causing any additional battery drain. I was at 100% at 9am, it's now 1pm and 93% with phone and data usage and downloading of those reinstalled apps on wifi. Seems to be on par with a normal day back on 7.0. I saw no major battery issue throughout yesterday either after updating and the problem being present.
Hope this info helps everyone with their own troubleshooting process.