Let $\{ e_r \}_{r>0}$ an uncountable orthonormal system in a Hilbert space $H$, prove that for every $v \in H$ $\langle v, e_r \rangle \neq 0$ for at most countably many $r>0$.
If we assume $\{ e_r \}_{r>0}$ is complete then because we are in a Hilbert space it means that $\{ e_r \}_{r>0}$ is a basis and hence for every $v \in H$ we have $v = \sum_{n=0}^{\infty}a_ie_i$ hence $\langle e_r,v \rangle \neq 0$ only for $r \in \mathbb{N}$ indeed.
How to go about the other case? tried in many ways but cannot find a proof
The fact that $\{e_r\}$ is orthonormal means that we can complete it to a basis $\{e_r\}\cup\{f_s\}$. Then since we are in a Hilbert space, your reasoning implies that $\langle e_r,v \rangle \neq 0$ or $\langle f_s,v \rangle \neq 0$ only for a countable number of indices $r,s$. But, since $\{e_r\}\subseteq\{e_r\}\cup\{f_s\}$, then in particular the number of indices $r$ for which $\langle e_r,v \rangle \neq 0$ is also countable.