If I have a C (70.76%) and get a zero on a test that is 10% of my grade, would I be passing or failing

Your average would drop to 63.68%. Ask your grader if that is passing.

Step-by-step explanation:

If 10% of your grade is zero, the remaining amount is 90% of 70.76%, ...

0.90·70.76% = 63.684% ≈ 63.68%

The letter grade associated with this depends on class or school standards. Ask your grader what it is.