If it's a new tank set up, then betas and neon tetras are not the fish to begin with. Both prefer aged water and preferably in the 78 degree range temperature wise. Sounds like you overwhelmed the filter with too many fish too quickly.
I've kept betas with great success in community tanks and have even had them spawn in those tanks, but they need some floating plant cover to feel comfortable in a community tank.
A 10 gallon tank is pretty small, so don't put too many fish in it. Try a threesome of hardy fish like platys or guppies, maybe a couple of cory catfish for some bottom interest.