No. Ballots are secret, so no one knows which offices you have voted for. Thus – while we can’t give tax advice – voting only for federal offices when you’ve received a full ballot should have no effect your state income tax liability.
Voting in state and local elections is hugely important: These races decide how your state will execute both state and federal laws on vital issues such as education, healthcare, and the environment. Moreover, state legislators and governors control re-districting for both state and federal offices following the Census.