I've read a topic about this before, and while no one had any proof that it would be healthy, there doesn't seem to be any reason why it would be unhealthy.

There were two things brought up, however. One is that the lucky bamboo shouldn't be submerged, and the second is that the plant will need to be rinsed very thoroughly before put in with the betta. Plants not intended for use in fish tanks can have chemicals on them.

I don't really know anything about lucky bamboo, so I can't tell you anything other than what I heard, but I hope it helps. Maybe ask at a pet store?

One thing suggested to me about trying out new tank decorations is to put them in a tank with two or three feeder goldfish for a week. If the fish live, the object is probably fine. I don't really love the idea, myself, but it is a way to find out.

They talk about this topic in more depth at the lucky bamboo forum.

I have a bowl with lucky bamboo and a betta, crab and a few neons.. so far only the neons die, not sure why. I do not fertilize the bamboo and it grows in sand so it never really has any growth to it, it stays about the same year to year.

Wow, I'm really late to this particular forum, but I have to tell you I have had GREAT success with bamboo and betta fish.
On my desk at work I placed a fish bowl. In the middle of the bowl I placed a short juice glass about 4 inches in diameter and placed three bamboo stalks in it and pebbles to anchor it upright in the glass. I then filled the bowl with spring water to within an inch or two of the top,completely submerging the glass, then added my betta fish. The water stays much cleaner (benefit of plant to fish) and the plant grows rich and green with no yellowing (benefit of fish to plant). It's a great symbiotic relationship and everyone comments on how beautiful the whole set up is. They like the crystal clear water and the very green bamboo.
Nina

Can lucky bamboo be submerged or grown in a 10gallon fish tank, What are good fish to begin with also?

As "lucky bamboo" is not actually bamboo, you may have better luck posting in the Lucky Bamboo forum.