If my opponent uses Unyielding Spirit and they have an Elder Dragon at base, then they play Fury of Blades, will the dragon ability still work?

To answer your question: No, the Elder Dragon’s ability will not kill your units in this scenario.

Ruling

Because Unyielding Spirit prevents all spell and ability damage, the damage from Flurry of Blades is prevented. Since no damage is actually dealt, the Elder Dragon’s ability (which modifies the lethality threshold of your damage) never triggers or applies to your units.

Reasoning

1. Prevention Replacement: According to Rule 437.4 and RiftJudge FAQ #9396, damage dealt to a unit that has all of that damage prevented is not considered to have been dealt at all.
2. Elder Dragon's Condition: Elder Dragon’s ability relies on the event of "your damage" being dealt to enemy units. Because Unyielding Spirit replaces the damage event with a "Prevent" action, the game treats the event as if no damage occurred.
3. No Damage, No Lethality: Since the damage is prevented and treated as 0 (Rule 437.2.a), the Elder Dragon's passive ability—which changes the lethality threshold for "your damage"—has no damage to modify. Therefore, your units survive the Flurry of Blades.

Note: This is my interpretation based on the mechanics of damage prevention and the Elder Dragon's passive ability. Official FAQs #9396 and #9918 confirm that prevented damage does not satisfy the requirements for Elder Dragon's passive ability.