Bungie recently clarified they will not remove any bounties from the players' inventory but they will not be as effective as one might've hoped.
Iron Banner bounties are one of the sources of Pinnacle weapons, meaning their Power can go all the way to the hard cap for the season, which makes them perfect for endgame content. It's for this reason that the Guardians would complete a bunch of them but not turn them in immediately. When the next season starts, they turn the bounties in and get a head start on the season pass as well as the race to the hard cap.
Hoarding the bounties became a point of discussion in the Destiny 2 community ahead of the Witch Queen release and Bungie stirred the pot by mistakenly stating completed bounties would be removed at the end of Season 15. They clarified their stance in the most recent TWAB and the TL;DR version is that the Iron Banana bounties will not be removed but they won't be as effective.
It's assumed they will still give the usual XP bumps but the bounties will not provide any Power advantage. Bungie singled out Iron Banner bounties and said the ones from S15 will be capped at 1350 Power, which is also the floor in the Witch Queen. In other words, any world drop at that point will have higher Power than the "pinnacles" you would get from hoarded IB bounties.
There is one set of bounties that will be removed, however - the ones from Tangled Shore. The zone is getting vaulted at the end of Season 15 so you might want to turn these in before February 22, 2022.