MOSCOW—Russia is exploring a raft of potentially unpopular new tax measures in the coming years, government officials say, as the oil-dependent country runs out of ways to plug holes in the budget.

Several Russian officials say the government is considering raising income-tax levels and increasing value-added tax but that any changes would only take place after 2018—a presidential election year. Any increase would be a sensitive issue in Russia, where real incomes shrank 9.5% in 2015 and the number of those living below the...