When do the research quests end?

There's no time limit on either type of quest. Do them at whatever pace you want.

Research quests are a permanent addition to the game, like raids.

The only time limit we know is that you can receive one stamp per day.

Other than that, it hasn't been around long enough for us to know if there is any sort of time limit for the day seven rewards.

And the common expectation is that sometime down the line, a new special research quest will be added (Celebi, perhaps?). It is unclear whether Mew will still be available to players who have not started the quest at that point (although I think consensus is that if you've started it, you'll be allowed to finish it. Note that all of this is uninformed speculation from people with experience playing the game but no inside information.