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 "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.
