The T-Mobile version of the LG G3 isn't bootloader locked, but all the others are. That's par for the course, but now you can finally do something about it. Bump! is a new bootloader unlock tool that will allow you to run a full version of TWRP and flash anything that has been "Bump'd."
The installation process is straightforward for anyone who's familiar with ADB. It's just a few commands, but you do need to be rooted ahead of time. Just use any of the root methods for the LG G3—doesn't matter how you get there. Check out the XDA thread for the full instructions, along with any updates added by the developers. The current builds of TWRP needed for the process are linked here.
Most of the ROMs you'd be interested in using on the LG G3 will be Bump'd, so there's nothing to worry about on the user side. The developers will post instructions on preparing Bump'd images soon, but it apparently isn't difficult. You can also keep up with root and Bump compatibility using this handy Google spreadsheet. As with all things of this nature, do please be careful. There are now pages on the TWRP site for all the G3 devices, but no official builds are up there just yet (just the Bump ones). We'll just link them here so you can check back later for an official build (assuming that's compatible with Bump!).
- TWRP for LG G3 AT&T USA [d850]
- TWRP for LG G3 Canada Bell, Rogers [d852]
- TWRP for LG G3 Canada Wind, Videotron, Sasktel [d852g]
- TWRP for LG G3 Europe, International [d855]
- TWRP for LG G3 Korea [f400]
- TWRP for LG G3 Sprint USA [ls990]
- TWRP for LG G3 T-Mobile USA [D851] (not new)
- TWRP for LG G3 Verizon USA [vs985]