I had two issues, first being the little indicator of new orders for orders that were 2+ years old that were partially refunded, partially fulfilled or canceled. Got rid of that by archiving those orders. The second issue with old partially fulfilled orders or canceled orders showing up in the default filters was "fixed" by creating my own filter adding the status for "unfulfilled and partially fulfilled" AND "open orders". You can alternatively delete the default "unfulfilled and partially fulfilled" filter thats there and just create your own with the same name including the "open orders" status.
Thought I'd share. I just had a discussion with our rep at Shopify about this problem and he said that all unfulfilled orders are supposed to be marked as unfulfilled. Even when it is cancelled, or an exchange order is made, the original order was never fulfilled and hence is supposed to always be marked as unfulfilled. So sadly I dont think Shopify has any plans of fixing this or the ambiguity it creates anytime soon.
Shopify fails to understand that unfulfilled is not a 'class' of order but is actually an important ecommerce task - like the most important todo item.
It's hard naming stuff sometimes but it's a mistake for Shopify to try and tell us that 'fulfilled' and 'unfulfilled' don't mean what everyone knows them to mean.
This is all so obviously silly that it's embarrassing sorry.