Contractor paying subcontractor

If I have a contract with a client, and hire a subcontractor, then submit his/her hours to the client, receive the payment, and then pay that full amount to the subcontractor, do I need to pay taxes on that?

The money you received is counted as income for your business. The money you paid to the sub-contractor is an expense (e.g. 1040 Sched C, Line 11). You likely owe the sub-contractor a 1099 and you're late if you didn't already send it.

The short answer is it depends on where you are located and the services performed.

For example in Delaware (since many have Del. corporations) there is a gross receipt tax :

Delaware does not impose a state or local sales tax, but does impose a
gross receipts tax on the seller of goods (tangible or otherwise) or
provider of services in the state. Unless otherwise specified by
statute, the term "gross receipts" comprises the total receipts of a
business received from goods sold and services rendered in the State.
There are no deductions for the cost of goods or property sold, labor
costs, interest expense, discount paid, delivery costs, state or
federal taxes, or any other expenses allowed.

Business and occupational gross receipts tax rates range from 0.1037%
to 2.0736%, depending on the business activity. In instances where a
taxpayer derives income from more than one type of activity, separate
gross receipts tax reporting is required. The type of business
activity additionally determines whether gross receipts tax is
remitted monthly or quarterly.