Strictly speaking, yes, the creature would still require a Fortitude save to negate the secondary effect.
The closest the rules get to discussing this case is
Whenever damage reduction completely negates the damage from an attack, it also negates most special effects that accompany the attack, such as injury type poison, a monk’s stunning, and injury type disease.
You’ll note that this is discussing damage reduction, not resistance, and that it’s talking about special effects that “accompany” the attack, attack riders so to speak.
For resistance, we have a slightly similar line, but, well:
When resistance completely negates the damage from an energy attack, the attack does not disrupt a spell.
Rules Compendium updates this wording to say specifically that it doesn’t force Concentration checks.
So nothing says that resistance can block effects that accompany energy damage. Moreover, unlike an attack with a rider, burning blood is one full effect, damage and action limitations as one thing.
All that said, burning blood explicitly describes the limitation of the victim to one move action as its turn as being due to “searing pain,” which makes no sense if the fire and acid aren’t actually painful to the subject. A successful Fortitude save eliminates the damage and the action limitation (for that round), which further shows that they’re supposed to go together.
So even though nothing says taking 0 damage negates the action limitation, and under a strict rules-as-written interpretation it therefore wouldn’t, personally I would absolutely waive the action limitation as well as the damage. As long as at least 1 point of acid and/or fire damage is taken, it would be in play, but if both are both literally reduced to zero, I wouldn’t apply the action limitation.