Can an infinite dimensional hilbert space have a finite orthonormal basis? Let $H$ be a Hilbert space with infinite dimension. Is it possible that there is a finite orthonormal basis $O$ for $H$, i.e. a set whose linear span is dense in $H$? What if $H$ is separable? 

This is no homework. Intuitively, I think it is not possible, but could not prove it.
No. The span of finitely many elements is finite-dimensional and therefore already closed.
As already stated by Andre S., it is not possible that a Hilbert space of infinite dimension contains a finite orthonormal basis. Moreover, an infinite-dimensional Hilbert space is separable if and only if it has a (then necessarily countable) orthonormal basis, see https://en.wikipedia.org/wiki/Hilbert_space#Separable_spaces.