Burning the Bootloader on Adafruit Feather 32u4
"Burn Bootloader" only works if you have a programmer connected to the device. The onboard bootloader in your Feather cannot change the fuses, even though the process may indicate it has done so.

You would have to connect an ICSP (In-Circuit Serial Programmer) to the correct pins (MOSI, MISO, SCK, RESET, VCC, GND). You can use another Arduino as a programmer by uploading a specific sketch to the Arduino to make it an Arduino as ISP.

The normal bootloader on an Arduino emulates this kind of device to allow programming by serial/USB port, but it has limitations. The ICSP process is the native way to get a program onto the AVR chip, and is the only way to set fuses.

EDIT:

This AVR USB Bootloader Datasheet states on page 3:


  • Boot Reset Fuse The Boot Reset Fuse (BOOTRST) can be programmed so
  that the Reset Vector points to the Boot Flash section start address
  after reset. Once the user code is loaded, a bootloader command
  (“start application”) can start executing the application code. Note
  that the fuses cannot be changed by the MCU itself. This means that
  once the Boot Reset Fuse is programmed, the Reset Vector will always
  point to the Bootloader Reset and the fuse can only be changed through
  the serial or parallel programming interface. The BOOTRST fuse is not
  active in the default factory configuration.


That document also references AVR282: “USB Firmware Upgrade for AT90USB” (Note, I could not find this on the Atmel website.) This document details the FLIP program, at Atmel tool for uploading firmware over the USB DFU interface and setting configuration settings, etc.

This doc has a FAQ section which states:


  
  Can I modify the fuse bits using Flip? • No, Flip cannot modify the fuse bits. To modify the fuse bit you can use either the JTAG ICE
  MKII, the AVRISP MKII, or parallel programming
  


So, while this may not be a definitive answer, it appears at this time that "burning fuses" cannot be done over USB on the 32u4. If that's not the case, I'm sure the site would welcome the new info.