The carabiner itself is of course problem-free. The eyelet is simply part of the silicone casing. You can easily cut it away with a sharp knife if you really don't need it. However, once it's gone, it's gone :)
I simply removed the carabiner, as it is too big for me, and replaced it with a very small mini hook, which is connected to the eyelet by a cord. This allows me to attach the whole thing directly to my trousers in a more manageable way.