How would the interaction between Den of the Wolf (partial: Characters lose all immunities) and the Triggered Effect on Burning Sword (partial "gains...immune to card effects") work? Would the Burning Sword character be immune to the Den's ability to shut down the newly gained immunity? I want to say that the BS effect takes precedence, but can't figure out why, so I might be wrong.

Until these cards are in wider circulation, full text would be appreciated.

Conceptually, there are two modifiers trying to act on a character; Den of the Wolf (which is "character loses ALL immunity") and Burning Sword (which is "character is immune to CARD effects," or essentially "character is immune to Den of the Wolf" in this example). Normally, when you have two things trying to modify the same card at the same time, you add the effects together and modify the card with the result. (Which is why +2 STR and -1 STR on the same character effectively becomes +1 STR.)

The problem here is that if one modifier is applied, the other one is not. So you need the result of the "immunity check" in order to see which modifiers should go into the "immunity check." It is a true paradox under the current rules.

There is no rule on whether the "gains" or "loses" modifier takes precedence. You could very clearly argue for either one if you tried. So until FFG comes out with a direct ruling, this is essentially a timing conflict (which modifier is factored into the check "first" dictates the outcome of the check). The First Player decides the outcome of such conflicts. So for now, I'd say the First Player decides which effect applies first and thus trumps the other.