If I have Doran's Shield equipped to a unit and my opponent uses Turn to Dust, will my Doran's Shield be killed during the beginning phase due to temporary?

Yes, your Doran’s Shield will be killed during your Beginning Phase.

According to Rules FAQ Clarification 2026-04-29 (and supported by RiftJudge FAQ #10676), while an attached gear's printed rules text (like its Tank ability) becomes inactive, any granted keywords remain active. Because Temporary was granted to your Doran's Shield by the spell Turn to Dust, that ability remains active even while the gear is attached. Consequently, the granted Temporary ability will trigger at the start of your next Beginning Phase and kill the gear.