When Henry's mother (Amy Morton) is celebrating Henry scoring a run after being walked, she hits her head on the lamp hanging above her (at around 57 mins). Morton obviously didn't mean to do this: she is visibly hurt and after sitting down mutters "oh... shit" (at 57:46, which is dubbed out on the audio, you have to lip-read).
When Henry has to take his first AB (against the L.A. Dodgers), the pitcher in that scene (Tregoraw) is Tim Stoddard. Stoddard was a technical adviser for the baseball scenes in the movie. Coincidentally, Stoddard was actually a pitcher for the Cubs in real life, he pitched on the '84 Cubs team that won the NL East title (and eventually lost to the San Diego Padres in the NLCS).
When Henry first goes to Wrigley Field as a pitcher, he goes to the players entrance. When he knocks on the door to be let in, an old man pokes his head through a hole in the door. At first he doesn't let Henry in, then Henry reveals who he is, and the old man says, "Well that's a horse of a different color," which is the same thing that was said in The Wizard of Oz (1939) when Dorothy and the gang get to Oz.
The scene featuring live shots of the crowd at Wrigley Field chanting "We want Henry" was filmed in Sept. of 1992, between games during a double header between the Cubs and their division rivals, the St. Louis Cardinals. The film's producers described the scene and instructed the crowd on what to chant over the stadium's PA system. While the majority of the fans participated, a large number of Cardinals fans in the crowd went off-script, chanting instead, "We want Ozzie"; referring to the Cardinals' all-star shortstop, Ozzie Smith. It took multiple takes for the entire crowd to participate in the correct chant, and the Ozzie chants are not audible in the film.
Filming for the Little League scenes took place at the Little League fields at the park district in Westmont, Illinois, which is a suburb of Chicago. Some locals were used as extras in the stands on the field.
23 years after the film's release, the Chicago Cubs would go on to win the 2016 World Series in seven games against the Cleveland Indians, coming back from a 3-1 series deficit to win the series 4-3 (including a 8-7 win in 10 innings in Game 7 at Progressive Field in Cleveland). It marked the Cubs' first World Series championship 108 years. Thomas Ian Nicholas celebrated the win on Twitter by posting the final shot from the film, showing Henry holding up his Cubs World Series ring to the camera.
Following the film's release, Thomas Ian Nicholas threw out the first pitch at multiple Cubs games and was also invited to sing "Take Me Out To The Ballgame" multiple times during the customary 7th-inning stretch at Wrigley Field. During the 2015 National League Championship Series where the Cubs faced the Mets as they did in the movie, he attended Game 4 in a Rowengartner #1 jersey similar to what he wore during the film.