The Shetland pony Tucker (Dustin Hoffman) had to be painted brown during filming, because his real coat turned grey. Everyone was so surprised, but the animal trainers did a little touch-up with brown dye.
The trivia item below may give away important plot points.
Reggie (Jeff Foxworthy) states "If you build it, they will come" while standing in the corn field. This alludes to Field of Dreams (1989), in which the main character chases a dream that no one believes in, same as Stripes is chasing his dream to race. The actual quote from Field of Dreams (1989) is, "If you build it, he will come."