I have a set Dust Tornado. My opponent has 3 Gem Beasts as Continuous Spel Cards. They draw and Activate the Field Spell card for the set- Rainbow something or another and I chain Dust Tornado. They said you can't destroy it since it says on it that they just have to have one on the field. That's only true if it resolves though, correct? Such as if I had no response, then later tried to dust it.

Ancient City - Rainbow Ruin must still resolve before any of its effects will become active. Destroying it in a chain with Dust Tornado prevents it from resolving properly, and it would it would not remain on the field at the end of the chain.

I thought so, but just figured I'd ask in case this card's effect prevented the normal rules.

when a chain happens, everything resolves backwards, so yea you're right.