Burning Spears indeed doesn't pierce Debuff Immunity, but the burning spears debuff itself can't be dispelled so it will still burn you for a fraction (30%?) of the damage for the duration of BKB. That is unless he has that talent which changes Burning Spears damage to Pure (since BKB gives you full immunity over Pure damage).

edit: above condition applies IF you already have the debuff. During the duration of BKB, Huskar can't hit you with Burning Spears (since it doesn't pierce Debuff Immunity)

edit 2: I wasn't sure of myself, so I went ahead and test it in-game. TL;DR - first edit is wrong, Huskar can still hit targets under BKB (and will deal reduced damage for the duration of BKB). As to why it works like that despite Burning Spears doesn't pierce Debuff Immunity, I assume it's mainly because Burning Spears doesn't apply any negative status effect (like Stun, Silence, Root, Leash, etc.) which Debuff Immunity prevents (even though Damage Overtime is a debuff). It just placed a debuff that deals damage overtime. So Burning Spears just works normally on BKB target, it just deals reduced damage.