Well, there's the fact that it was originally written as an animal rights tract for adults...
The animals provide the majority of the tear jerkers, but the human characters get some in too. Jerry Barker, left waiting out in the cold for hours while already sick and doing his best to care for Beauty the whole time, provides a lot of pathos, especially in the 1994 film adaptation.
Ginger's fate: After being ruined by a careless rider, her 'breath' (essentially, her trachea) is strained and she is sold down lower and lower because of her subsequent breathing difficulties, to the point that when Black Beauty sees her again, she's so broken down he doesn't immediately recognize her. All her fire and spirit that she had at the beginning is gone, and she says she wishes she would drop dead in her harness and be out of her misery. Shortly after this, Beauty sees what looks like her body...
Oh! If men were more merciful, they would shoot us before we came to such misery.
The really heartwrenching (mixed with Fridge Horror) part is that we're not even completely sure if the dead horse Black Beauty saw was actually Ginger (it could've just been another horse that happened to look like her), but Black Beauty sincerely hopes that it was. All Black Beauty saw was a chestnut horse with a long neck and white on the face, but these are by no means rare features among horses, and it's possible that Ginger spent many months in agony and suffering, ending in a trip to the slaughterhouse.
The terrier puppies in the anecdote Sir Oliver told the other horses, about how the people cut off their soft, floppy ears. It took another hundred years, but finally ear-docking was outlawed in the United Kingdom, and is no longer being taught in veterinary schools in the United States.
Black Beauty being injured because of Ruben Smith's drunkenness, and Smith's death, leaving his wife and children alone. YMMV on how sad Smith's own death was, because he was drunk so it was partly his own fault, but it was said that Smith was generally a good man, and well-liked.
Following on from that, Black Beauty standing there with lacerated knees and a badly stone-bruised hoof, listening to Smith die, praying for someone to come, and eventually simply thinking of his mother and childhood.
Beauty and Joe's reunion in the 1994 film (also counts as heartwarming). Joe's reaction when he pushes Beauty's bangs away and recognizes the star on his head...
For added effect, go to about 2:55 on this soundtrack which plays during this part, and try to hold back the tears.
Also Fridge Horror, but at one point in the story Black Beauty sees a pony that looks exactly like Merrylegs being whipped until he bled and having his mouth torn by the bit. Again, Black Beauty hopes that it is not Merrylegs, but never states this for certain.
It probably wasn't, because when Merrylegs was sold to the Vicar the conditions were that he would be cared for and never sold, and when he got too old he would be shot and buried. Of course, the Vicar may not have kept that promise...
The Fridge Horror is further fueled when you remember Beauty and Ginger were sold to the Earl of W—- because he was a friend of Squire Gordon, who hoped they would be treated well on account said friendship. Before the year was out, both were ruined (Beauty's knees and Ginger's trachea) and pawned off like dresses going out of season. While not malicious, the earl didn't make sure they were properly looked after in his home any more than he made sure they got good homes. He even laments the money he paid for them more than his disservice to a friend. Who's to say the vicar did any better? Or that something didn't happen to him (like with Jerry), and the people around him didn't honor the promise?
In the book, Beauty initially shares the role of pulling Jerry's cab with Captain, an Old Soldier who served as a cavalry mount during the Crimean War. He and Beauty strike up a fast friendship, but then a drunk rams his cart into Jerry's cab and poor Captain is seriously injured. Jerry can't afford to keep a horse who can't work, and he's far too aware that most of the people who'd buy an old and invalid horse like Captain would probably treat him horribly. Heartbroken, Jerry comes to the conclusion that "the kindest thing he could do for the fine old fellow would be to put a sure bullet through his heart."
The last scene in the book, where although Black Beauty is safe and happy, he chooses to dream of his days in Birtwick, standing with his old friends in the orchard. The fact that he never even saw all but one of those friends again, much less say goodbye to them, is somewhat painful when you consider it is very likely that Black Beauty is the only one still alive and the world he lives in is generally not kind to horses.
The gorgeous 1994 film is absolutely heartwrenching from start to finish. Most notable are the scenes where Beauty, Ginger, and Merrylegs spend their last day together (especially the dance), Beauty reunites with a broken and beaten Ginger and is almost glad when she dies because her suffering is ended, Joe finally finding Beauty at the horse fair and openly weeping over the little white "star" mark, and Beauty daydreaming that he is with his two friends again.
The credits (with that beautiful, haunting music playing) shows Beauty, Ginger, and Merrylegs cantering and rearing and bucking, looking for all the world like children playing tag. Anyone who's ever loved and lost a horse can't not cry when they watch that.
At the start of the book, a Beauty and his mother, Duchess, witness a hunting party chasing a hare. One of the horses is injured too badly to recover and is given a Mercy Killing. Duchess is deeply disturbed by this, not because of the shooting, but because she was very familiar with the horse in question, whose name was Rob Roy. Afterwards she avoids the area of the meadow where she saw the event. It isn't until several chapters later that we learn Rob Roy was actually her son, making him Black Beauty's older brother.
This quote that describes how it is for every pet animal out there:
"We don't get to choose the people in our lives. For us it's all chance."