If I use Zhonya's on my Temporary unit, will it die on the turn after Zhonya's got triggered?

Yes. The unit with Temporary that was saved by Zhonya's Hourglass will die at the start of the next turn's Beginning Phase.

Here's the sequence:

1. Temporary triggers at the start of your Beginning Phase, attempting to kill the unit.
2. Zhonya's Hourglass (a replacement effect) intercepts that death — it heals, exhausts, and recalls the unit instead.
3. However, the Temporary keyword remains on the unit. Saving it from one trigger does not remove the status.
4. At the start of your next Beginning Phase, Temporary triggers again and this time kills the unit unless another effect saves it.

Temporary is a trigger that happens once per Beginning Phase. Zhonya's replaces one instance of death, but the keyword persists, so the next Beginning Phase it fires again (FAQs #301, #5544, #8940; Rule 816.1.b).