I am watching the Mission Impossible movies for the first time, and it appears to me that if you skipped Mission: Impossible 2 there would be no problems. The love interest doesn't seem to be mentioned in later movies, and I can't think of any other story elements that are affected. Am I missing anything?
In Mission Impossible 2, the ONLY thing that is continued or relevant to MI series is, assigning a mission to Agent Hunt and his helper Luther Stickell.
All other things including Characters, Villian, Scenes or girlfriend. (wife is not mentioned at all) are different. So theoretically, it's an independent script of the series.
Some of us will say, MI:3 is taking a bit of idea from MI:2 and that is Rabit's foot, which resembles with the biochemical weapon (Chimera) in MI2. But then in MI:3 also, It's not revealed clearly what exactly the Rabit's foot is. So we can't come to conclusion on the similarity between that and a biochemical weapon. Such Biochemical attack abandoned by IMF at the beginning of Mission Impossible: Rogue Nation as well.
Certainly, these all are after MI-2. So, only an idea of a chemical weapon doesn't demand knowledge of the whole series.
The Mission impossible movies are meant to function essentially like the Bond movies, with the main plot of each story being self contained. The only thing that really continues from movie to movie are the ensemble characters and their roles, such as Simon Pegg or Jeremy Renner who joining in MI:III. However you could ignore the movies and still generally get what these characters roles are in the team as they fill an archetype (Pegg is the tech guy). Though it does look like the next movie MI:Fallout will continue on from Rogue Nation.