Symbolism, Imagery, Allegory
Sometimes a cigar is just a cigar, but this tree? We're pretty sure she's not that simple.
She, Her, Hers
For one thing, if Shel Silverstein had wanted the tree to be "just a tree," he probably wouldn't have given her a feminine pronoun. "It" would have worked just fine for a tree that wasn't supposed to be anything more, but this tree's pronouns are she, her, and hers, and we think that's significant.
With a feminine pronoun, the tree's symbolic possibilities increase. The tree could be a symbol for a mother, lending credence to the idea that the story is about parental (specifically maternal) love and the sacrifices parents are willing to make for their children.
Assigning the tree a feminine pronoun also helps to paint her as human, something "it" wouldn't accomplish. This creates the potential for the tree to represent one half of a codependent human relationship in which one person takes while the other gives. This is how Silverstein himself characterized the story in a 1978 interview.
Ignore the Man Behind the Curtain
But just because Silverstein said it doesn't make it absolute. For one thing, he hated doing interviews, so his answers were occasionally flippant and abrupt. And for another, he also summed up the story on a separate occasion by saying, "It's the story of a boy and a tree. It has a pretty sad ending" (source). So…onward we go.
More About the Pronouns
The feminine pronoun also works well if we want to equate the tree with nature. (Ahem, we're pretty sure Mother Nature is a woman.) If we choose this interpretation, the story becomes a cautionary tale about humanity's poor treatment of the environment.
The boy (humanity) carelessly consumes the tree (nature) without producing anything of lasting significance. And once the tree is reduced to a stump, she's not coming back. That means no more fruit, no more lumber, and no more conversion of carbon dioxide to oxygen. Sorry, humanity. Your loss.
Of course, this reading of the book is possible even without the feminine pronoun. After all, trees are essential to human life. We couldn't breathe the Earth's air without them (unless a whole lot of other plants sprang up to take their place). So the pronoun isn't necessary for an environmental reading of the book, although—considering the Mother Nature connection—calling the tree "she" doesn't hurt.