If I conquer a Minefield and trigger my Ivern legend, will I still have to burn 2?

Yes — you still burn 2.

Minefield's "When you conquer here" ability triggers the moment you conquer, and so does Green Father's. Both go on the Chain as independent triggered abilities, and once Minefield's trigger is on the Chain it resolves regardless of whether Minefield is still in play (FAQ #9525 / #10867). Replacing it with a Brush token before its trigger resolves does not stop it.

You choose the order both triggers are placed on the Chain (Rule 383.3.d), but either way Minefield's burn 2 still happens.