These come from the IMDB Enjoy!

Revealing mistakes:
When Hopkins boards the City of Paris, men can be seen climbing rigging in the background. However, the rigging they are climbing does not go all the way to the top of the tower that supports it, so they are essentially climbing to nowhere.

At the end off the movie when Frank releases Hidalgo back to the wild it still has its steel shoes on (he would know that he had to remove them to prevent damage to the horse's hooves).

Factual errors:
Jazira said she was going to become the prince's fifth wife. Islam permits a man a maximum of four wives.

Audio/visual unsynchronized:
When Frank Hopkins is giving water to a dehydrated traveler, he holds a flask to his mouth. Clearly, no water comes out, yet the sound effect was added.

When offered a smoke by Sheikh Riyadh, Frank takes out a cigarette, which remains unlit until he takes two steps away, and it becomes lit and half smoked.

When Frank is racing to the finish at the end of the movie, supposedly riding Hildalgo bareback, a saddle appears and disappears between shots.

The blood on Hidalgo's nose as he approaches the finish line disappears for one shot.

When Frank Hopkins is applying shaving cream to himself, he spreads it all around his chin. In the next shot, we see it covering only half of his chin.

Errors in geography:
When Hopkins leaves New York on the ship and passes the Statue of Liberty, a mountain range is visible in the background.

Damascus is the finish line for the race. In the movie, the finish line is just yards from the ocean. Damascus is 60 miles inland.