The answer differs for federal elections and state/local elections.
For federal elections, there is NO link. Federal elections include elections for President and Vice President, the U.S. Senate and the U.S. House of Representatives. An easy distinction is - if you're voting to send the candidate to Washington, it's a federal election.
A. Federal: If you vote for federal offices only, the act of voting will have no effect on your liability to pay State income tax or any other tax. You should have no hesitation about voting in federal elections, regardless of your state and/or local tax status, as for federal elections there is no link between voting and taxes.
B. State: If you vote for state or local offices, under state law, the act of voting could conceivably be used to establish your domicile and therefore your liability to pay State income tax.
The U.S. Government's Federal Voting Assistance Program clearly states:
Keep in mind that exercising your right to vote in elections for Federal offices only does not affect the determination of residence or domicile for purposes of any tax imposed under Federal, state or local law. Voting in an election for Federal office only may not be used as the sole basis to determine residency for the purposes of imposing state and local taxes.