Neem oil has a very bitter taste, so it is not something that one would want to eat. In India, they do consume neem oil for medicinal purposes. However, this can be dangerous for young children.
Generally, there is not enough information to determine whether consuming neem oil is wholly safe. In the UK, consumption is not recommended due to a lack of safety testing.
So, Can I use neem oil to clean my teeth?
There is good evidence to suggest that neem oil can contribute to healthy teeth and gums, as well as help with bad breath. Although it is untested in the UK.
You can use it by applying one or two drops to your toothbrush and cleaning as normal. This will allow the neem oil to get between your teeth and into your gums. Though please see above regarding the safety of consuming neem oil !