New York Rangers - Mystery Alaska
San Diego Padres - Looks like we’re stuck with The Kid From Left Field. God, will someone please make a Tony Gwynn biopic?
And if you want to count defunct teams, there’s Damn Yankees for the Washington Senators.
Well, one has actors portraying historical events and the other relies exclusively on actual footage–a divide that’s pretty clear. Probably the film that has the most grey area between these two camps is The Jackie Robinson Story where the whole film is reenactments from history, but Jackie plays himself.
This film is better known to current moviegoers, but it’s still about the team from the outside looking in, while mine is the story of real life BoSox player Jimmy Piersall.
Brian’s Song is an extremely compelling reason to reevaluate the TV movie rule. So let’s! 
Well, since I included the Brooklyn Dodgers in my OP, the Senators are just fine.
And I remember the Gary Coleman film growing up, myself a SD resident and big Padre fan, but didn’t include it in the OP because of the TV movie rule–which has since been turned over. Good addition!
I originally excluded this because it was a TV movie, though I also think the Gehrig biopic–w/11 Oscar nominations–is still better known, though personally, I can’t think of a more famous baseball movie that dislikes baseball as much as it does (link is to my blog about 42 and PotY). So, I think the Maris/Mays film is significantly better and an excellent alternate.
The Boston Red Sox played a big part in Field of Dreams. That’s Oil Can Boyd pitching in the game that Ray Kinsella and Terence Mann attend.
In a very hilarious manner:
Green Bay Packers - Pitch Perfect 2
Also,
Miami Dolphins - Ace Ventura: Pet Detective
Yes, but it’s got more about the White Sox in it. (The New York Giants are referenced as well, in the person of Moonlight Graham.) In any case, both Soxes have other movies more directly about them.
But the movie doesn’t use the Jets, right? Any idea what movie might?
In that case, I resubmit Brian’s Song and Fighting Back, which was about Steelers’ running back Rocky Bleier.
I seriously doubt there have been a lot of movies made about the Arizona Cardinals, so by process of elimination I think they have to be represented by Jerry Maguire.
Paper Lion. Detroit Lions
If by that you mean that the North Stars became the Ducks, that’s incorrect. The North Stars relocated to Dallas, as the Dallas Stars, for the '93-'94 season. The Anaheim Mighty Ducks were an expansion franchise (not a relocation) in that same season.
No, Pittburgh Steelers = “The Steeler and the Pittsburgh Kid” from the Mean Joe Greene commercial 
Detroit Tigers - Tigertown (I remember that on the Disney Channel)
I have never heard of that movie, and I doubt anybody else has, either, but that commercial is so iconic, I won’t challenge your choice.
Detroit Pistons: “Bad as I wanna be”
No, I meant that the movie is the signature of the Minnesota North Stars. The Anaheim Ducks did not come into existence until after the movie was released. Yes, I know that the Stars moved to Dallas, I probably know more about that than just about anyone.
The Mighty Ducks movie did feature Gordon Bombay talking to North Stars/ Stars icon Mike Modano and long time fan favourite Basil McRea.
OK, friends on my Facebook added some football suggestions so here’s a full NFL listing so far (bold are additions to the thread):
Arizona Cardinals -Jerry Maguire
Baltimore Ravens -The Blind Side
Buffalo Bills -Buffalo 66
Chicago Bears -Brian’s Song
Cleveland Browns -Draft Day
Dallas Cowboys -Black Sunday
Detroit Lions -Paper Lion
Green Bay Packers -Pitch Perfect 2
Los Angeles Rams -Heaven Can Wait
Miami Dolphins -Ace Ventura: Pet Detective
New England Patriots -Ted 2
New York Giants -Big Fan
New York Jets -Big Daddy
Philadelphia Eagles -Invincible
Pittsburgh Steelers -Fighting Back
San Francisco 49ers -There’s Something About Mary