I think you need the word 'that' to make the sentence completely grammatical, but I can't quote you the reasons why.
I'd put it like this:
"Our neighbour asked that visitors please not smoke in his house."
So you need both an explicit object (he, she, it, we, they, visitors) and the inclusion of the word 'that'. Better grammarians than me will be able to explain why 'that' is required!
Another way is to use an infinitive:
Our neighbour asked visitors please not to smoke in the house.
As noted by @human, it is also acceptable - and perhaps more natural - to say
Our neighbour asked visitors to please not smoke in the house.
on the basis that split infinitives are not a error anybody needs to take seriously.