Yes. Good-Til-Cancelled (GTC) orders that take multiple trading days to fill in their entirety will be charged separate commissions (plus any relevant per-contract charges) for every day on which part of the position is filled.
Modifying an order effectively cancels the original order and books a new order. Partial fills on the original order are subject to the commission at the time of the order. Commissions for fills for the modified order are calculated as if it were a new order.
Note that GTC orders remain open and working until cancelled by the customer or for 60 calendar days from order entry.