I'm hoping they do, but it would actually get tricky.

Law enforcement wants info on a phone number -> it goes to Twilio -> Twilio links this number at that time to Burner -> Burner links this number at that time to a user account. At this point, I'm not sure how they can link it to a real person. I'm guessing the credits are from in-app purchases, which means Burner itself might not have any real personal info on you and the buck then passes to Apple.

That means three different privacy policies involved and most likely a pretty long delay.

I'm sure Twilio can link calls to IP addresses and the phone companies can do the rest.

The IP address would be the Burner servers, not the end user.

Only if Burner is handling the outgoing phone calls (to your real phone number) themselves. Seems like you could bridge a phone call to a Twilio phone number to the end user's real phone with the <Dial> element pretty simply: http://www.twilio.com/docs/api/twiml/dial

You don't have to connect a call to your phone, you can just use Voicemail.