For those wondering, Mission Challenges in Call of Duty: Modern Warfare will not reset at the end of the seasons in Call of Duty: Modern Warfare.
Missions provide fans a way to unlock additional cosmetic weapon variants, alongside other loot and XP bonuses. There is one set of bonuses that were added to the game at launch, and all of those will remain available to use and activate even when Season 1 begins.
If you currently have an active mission challenge, they will remain active and allow players to continue to complete those challenges at their own pace.
Infinity Ward even confirmed this in tweet:
So if you’ve still got Missions left to complete, you will still be able to finish them!