They've always removed mine.

I have a habit of not removing the baggage tags once I pick it up, mainly because when I check into the hotel at destination, the hotel staff in 9 out of 10 cases will look at the bag tag and see that I came from X origin. Saves me having to answer a question because they usually ask otherwise.

And seeing that I'm usually always running behind schedule, I end up having to rush to the check in counter so all the stickers are still on my bag. I then don't end up seeing what happens to the bag as I head off to the gate but when the bag comes out, all of the old airline stickers (hotel ones stay) have been removed, with a new priority tags attached. When interconnecting to another airline, some airlines even bother to retag the bag while its in transit.