How can I sum values within a specific time and date range?

You'll need to use SUMPRODUCT instead of SUMIF since you need to satisfy multiple criteria/conditions for the sum.

Assuming the values in column A are dates and not strings, this is the formula you need:

=SUMPRODUCT((dates+times>=start_date+start_time)*values_to_add)-SUMPRODUCT((dates+times>end_date+end_time)*values_to_add)

Here it is in action, using the data range you provided as an example:

If you merge columns A & B, you'll only need a shorter formula than the one I previously mentioned. You could also use Excel's Advanced Filtering tool and then obtain a subtotal of the filtered cells.

You could add a subtotal cell at the end of column c, then create a table out of your data, and filter using only the dates you want. =sumif() is probably faster, but may be less flexible in the end.

This will work without an array formula, letting I1 be the earlier date, and J1 be the later date:

=SUMPRODUCT((A1:A6+B1:B6 >= I1)*(A1:A6+B1:B6 <= J1)*C1:C6)

If you want to use an array formula, then you can just use a SUM:

{=SUM(IF((A1:A6+B1:B6 >= I1)*(A1:A6+B1:B6 <= J1),C1:C6,0))}

but I'd never use an array if I didn't need to.