Are you limited to only receiving burn cards as challenge rewards, or is there someway to get them for just scoring? What happens when you run out of challenges?
You get burn cards everytime when:
you also receive cards randomly as you do pretty much anything in the game:
You will see "burn card earned [task]" (example "burn card earned grunt" for killing some grunts) in red/orange in your xp feed when you collect a new card.
The cards you get from challenges are more related to the challenge itself (Gun challenges will give you amped versions of that gun), while gameplay cards are completely random.
Yes, various actions (killing stuff, not noticed any for peaceful play) in game may randomly award you a burn card. Completing challenges is not the only way to acquire them, and from what I've played so far it isn't even the biggest provider of them.