In countries where you're obligated by law to vote, such as Australia and Belgium, one would be obligated to vote based on the concept "dina d'malchuta dina - the laws of the land are law." This means that a Jew is obligated to follow the laws of the country in which he lives.
In countries where voting is not obligatory by civil law, a Jew nevertheless has a responsibility to actively help in establishing a just society. As our Sages say in Ethics of the Fathers (2:3), "Pray for the peace of the kingdom (government) for if not for the fear of it, people would swallow each other alive." Voting is one way of helping establish a better society, and hence one has a responsibility to do so.
In Israel, where there are Jewish political parties which represent various Jewish interests, it is not unheard of that Rabbinical leaders issue rulings if there is a mitzvah to vote in a specific election, and if so, for which candidate(s).
Sources
- Bava Kama 113b
- Shulchan Aruch, Choshen Mishpat, 369:2
