Is it possible for certain infections in the finger to erode the bone?

Yes. Osteomyelitis - a medical term for bone infection - can lead to erosion of the bone by the bacteria and by the body's attempt to eradicate the bacteria. This does not always occur with osteomyelitis, though, and osteomyelitis does not always require surgery to cure. Oftentimes, it can be cured with IV antibiotics alone.