How does Burner activity look on a phone bill, and does it use wifi or data?

All calls inbound and outbound will display the burner number.

For instance, if your burner number is (333) 333-3333 and you get a call to that number from from (444) 555-1234, your phone bill will show a inbound call from (333) 333-3333 (not (444) 555-1234).  Likewise, if you make an outbound call from Burner to (555) 555-5555 your bill would show an outbound call to your Burner number, not the (555) number.

As for the person you call / who calls you, their phone bill will show your burner number too.

In both cases, your burner line bridges you to the caller and vice versa, so the number showing up on both phone bills will be the burner number.

Your text messages are different, in that they do not forward or bounce between your main cell and your Burner. SMS is sent and received directly through the app, and will not be on your carrier bill. 

Burner uses your phone's data but it can also run on Wifi. Below are the instances in which you might use Wifi vs. cellular data: 

Cellular Data
- Running the app, SMS, MMS, calling

Wifi
- Running the app, VoIP calls, SMS, MMS

In order to use the default call flow, your cellular data needs to be turned on so your Burner calls can forward to your main cell number. If it's not, your calls can't go anywhere. The exception to this when you have Wifi calling turned on for your main cell number, in which case it can receive Burner calls when you are on Wifi only and have no cellular data.