One of the most telling things that came out of the reports after Burke's firing, was how heavily involved he was in CBA negotiations.
I heard a few times that he was heavily involved in the cap-savings penalty/Luongo rule being implemented into the new CBA. Which makes absolute sense; especially when you consider just how dead against circumvention contracts Burke was under the old CBA.
It makes sense to me - he doesn't like cap circumvention deals, he worked hard to penalize teams that have those deals in this new CBA, and doesn't want Luongo who represents that very sort of contract.
While it might not have all been on Luongo, I think the loggerheads that the board and Burke found themselves in in regards to Luongo definitely had a huge influence on the decision to get rid of Burke.