Make it believable.
If the fairies are small, they might well be threatened by animals harmless to humans. For example, mice do not find cats adorable, and thumbsized fairies probably wouldn't either.
But then cats are predators and would indeed eat small fairies, while rabbits (as far as I know) nibble on grass, so how could they threaten the life or stuff of fairies? Maybe they eat the grass-made houses?
To make the threat by rabbits (or whatever you have in your story) believable, take in the perspective of those threatened by it. If you write about rabbits from your human perspective, the rabbits munching the fairy huts will be laughable. But if you get yourself into the fairies and write how they experience this threat, the humor will be gone.
Think of pigeons. Many people love pigeons on the cities. They find them adorable and feed them. Now think about them from the perspective of city officials, who have to battle the effect of pigeon droppings on buildings and monuments, and the public health office, trying to contain the potential of human infection from pigeon droppings. You can tell that story in two ways: either you start with the parents-and-kids view and slowly shift to the view of the officials, taking your readers, who presumably start on the pigeons-are-adorable side, with you to the pigeons-as-problem side; or you start with the problem-battling mindset, focussed on the problem and leaving no room for the naive adoration.
In a more adult, action, mystery and suspense focussed rabbit-battling fairy story I would probably choose the latter; in a children's tale focussed on the wonder and strangeness, the shift from our world and views to the fantastic is common and I would probably employ that.
As to your musings in the comments:
If you conceive of fairies as human-like beings living basically human-like lives, then it is very plausible that they do not only have to deal with supernatural enemies. In fact they are no longer supernatural at all, but just a different species, and they will of course have to deal with all the everyday problems that all other species have to deal with as well, including their small grass huts getting eaten by rabbits.
But if you coneive of fairies as supernatural beings, then humanizing them in that way is indeed breaking believability. Supernatural beings are not part of any biome. Trolls do not get bitten by insects, even though they live in the gnat infested forests of Scandinavia; selkies do not get eaten by orcas; and faries do not have bad hair days. These beings exist only in relation to human thoughts and emotions, not to the physical world, and any problems they might have are derived from this relationship. They may have to deal with sin or feel bothered by humans sleeping on their fairy hills, but there is (to my knowledge) not one single traditional tale where a mole digs up such a hill.
So be careful what you want your faries to be. Do you want to humanize them, as has been the fashion in recent fantasy? Or do you want to tell of them as the powerful mystery that they were once believed to be? You cannot have both.