Born Free

Gorillas in The Mist

Turner and Hooch

My Dog Skip

Benji

Old Yeller. I love that movie.

Every Which Way But Loose
Any Which Way You Can
Free Willy
Any Lassie movie
Ace Ventura Pet Detective

Broadening the definition of "animals" for some of these, but:

The Birds
Jaws
Cujo
Turtle Diary
Day of the Dolphin
and, my personal favorite, Air Bud :)

Babe

Homeward Bound