No, bettas will not eat neon tetras. Bettas and neon tetras can live together in harmony if they live in a supportive environment. If you want to be successful in keeping bettas and neon tetras together, make sure they have a big enough tank and plenty of plants to keep them happy.
Bettas can be aggressive, and neon tetras can be fin nippers if they are not properly looked after. However, you can maintain the peace by keeping your neon tetras in a big enough school and giving your bettas plenty of plants to hide in.
Make sure to also give your pets lots of swimming space.