If I play Turn to Dust on an attached Equipment, will the Equipment die in the Beginning Phase?

Yes. The granted Temporary keyword will be active. It will trigger and kill the equipment in its controller’s next Beginning Phase.

Attached Equipment only have their printed rules text made inactive per rules 718.2. and 135.4. This means that any granted rules text will still be active; granted Temporary will trigger and kill attached Equipment and granted Deflect will add [A] to the cost of enemy spells and abilities that target them. Prior to this rules update, all rules text of attached Equipment was inactive.

Rule 135.4.: A card’s printed Rules Text is Inactive while that card is Attached to another card. Rule 718.2.: While in this state, the card’s printed Rules Text is Inactive.