Scenario:
Lilith is within 1" of a Fire Gamin. Lilith has Burning +2 on her. The Fire Gamin has 1 Wd left. Lilith resolves the burning, takes 2 damage, triggering her Black Blood ability, damaging all models within 1". The Fire Gamin, being within an inch, takes damage and dies, triggering Flaming Demise. This causes 1 damage plus Burning +1 to all models within 1". So Lilith takes 1 damage (triggering her Black Blood again) and gains Burning +1.
Do the effects of that burning resolve during that turn's Upkeep or during the following turn's upkeep?