In principle the GST/HST is intended to be neutral to most registrants. (A registrant is one who is registered for GST/HST).
Unfortunately TurboTax is unable to comment on the specific mechanics of Uber Eats payments to drivers. However, the general GST/HST timing rule is: essentially, the tax is collectible by a registrant at the time a taxable service is provided, not later.
Regarding GST/HST on some expenses incurred in the course of providing a delivery service: a non-registrant can claim the GST/HST incurred as an expense for income tax purposes, which is less advantageous than the full GST/HST recovery that a registrant gets from claiming an input tax credit.
