Why is my belly button sore and irritated?

Umbilical discomfort. Your umbilicus (belly button) can be sore for numerous reasons, but the most common are: 1. You are washing it either too much, or 2. You are washing to too little. If, however, you are pregnant, then it is common for pressure to build up and push out the umbilicus, which can also cause discomfort. Lastly, my may have an umbilical hernia.
Could be fungal. Many reasons, it could be as simple as lack of regular hygiene of the area or could be due to a fungal infection. You can try one of the over the counter antifungals used for athlete's foot.