I've heard of many people successfully keeping neon tetras with bettas, in the proper environment.
First off, it depends on the betta. Some bettas are more aggressive than others. If you have an aggressive fish, that flares at everything that moves, I don't suggest it. But if you have a betta that seems very gentle you should be fine.
Second, you need a minimum of 10 gallons. Neon tetras are shoaling fish, and I believe the minimum number you need is 5 or 6.
Third, you NEED a cycled tank before adding any neons. Neons are very sensitive to water changes and can die very easily at a water param change. I bought several neon tetras and didn't cycle the tank (this was before I knew about cycling) and they all died within a day.
You should also have a tank with lots of coverage. This coverage is helpful to the neons, should they feel they need to escape if your betta begins chasing them. It is also helpful to your betta if the neons try to nip at your betta.
Remember, if you buy any tetras, PLEASE QUARENTINE THEM FIRST.
Even if a fish doesn't look sick, it could still be carrying an illness. Minimum QT time is 2 weeks.