I'm going to take a shot at this.
I remember when I was young there was an animated short on a VHS. That had a story about a puppy.
This has bugged me forever, because all I remember is near the end the puppy is injured (I think the puppy is laying by a fence in a field) and seems dead, and one of the characters (I think another animal) comments about "his nose not being cold" after checking to see if he's alive.
This is probably a lost cause because I don't provide a lot of info but if anyone has the slightest idea of what I'm talking about please respond. If not oh well.

Could it be Balto or All Dogs Go to Heaven (1 and 2), they're not really all puppy movies (or short films), but they're animated and from the 90's and the dog seems to die at some point or another.
Do you happen to remember what the animal was that spoke at the end it might help some.

It sounds like the movie Tiny: The Seventh Brother http://www.youtube.com/watch?v=2H-xXEsMwkc