There are many factors that can cause this, but the simple answer is that your plant just isn't happy. The tips of the leaves on a Peace Lily are often the first areas to show signs of distress.
The top reasons the tips of your leaves might be turning brown are:
1. Watering too much or too little.
2. Lack of humidity for the plant.
3. Too much fertilizer.
These are in order from most common to least common. First, double check that you are not over watering your plant and that it is in an area where humidity is high. A bathroom that often gets steamy is ideal but if that is not an option, consider placing the plant on a pebble plate filled with water. Misting the plant with warm water occasionally can also help to raise the humidity levels.
Lastly, if you are using fertilizer, ensure that you are not using too much. Peace Lilies can be sensitive to indoor plant fertilizers. If you are using a generic fertilizer for indoor plants, consider cutting the recommended dosage in half for the Peace Lily.