I was finally able to figure out how to fix my Powerbeats3 from constantly disconnecting from my phone. It was incredibly frustrating to see it disconnecting every 5-10 minutes or so. This is what worked for me...
1. Connect your Powerbeats3 to your computer and use the Beats Updater app to see if there's a firmware upgrade available. (Mine already were up to date)
2. Go to Settings > Bluetooth > Powerbeats 3 > Forget Device
3. Go to Settings > General > Reset > Reset Network Settings (this will reset your wifi passwords too but it's the only thing I could do to make it work)
4. Reset your iPhone
5. Reset your Powerbeats (Hold the power button + Volume down button at the same time until the red light flashes)
6. Repair your Powerbeats to your iPhone