It appears that newer builds of the Sling TV app are blocking unofficial devices such as Android TV boxes. Interestingly enough, sideloading the apk to the Amazon Fire TV also has stopped working. The Fire TV is advertised as coming soon on Sling's website, but I guess they want us to wait for an official Amazon App Store download.
This is only affecting new downloads. Older builds do seem to still work on these platforms and there hasn't yet been a forced update.
You can download build 4.0.3.137 here. This build is confirmed working, and you can watch it launch on a Fire TV stick as of this morning below: