While many consider the forbidding of interest (riba) to be a key component of being an Islamic economy, and Iran technically does this for its state-owned banks (although not in practice), what arguably makes it the most "Islamic" of economies is the large role in the economy of Islamic charity foundations known as "bonyads" (Persian for the Arabic "waqf"). Since the 1979 Islamic revolution, these have come to play an important role in the economy, controlling by best estimates on order of 20% of Iran's GDP, all of this being in the non-oil sector of the economy (http://en.wikipedia.org/wiki/Bonyad). The Bonyads do not pay taxes and do not have to publicly report their activities (and have been accused by many as the conduits for funding of foreign groups such as Hezbollah). Controlled by mullahs (Islamic clerics), they are supposed to be charitable organizations, but most appear to do little of this and to be corrupt and bloated institutions propped up by subsidies, with 80% of them reportedly losing money. They seem to be adding to inequality in the economy, and unhappiness with their role was part of what drove support for Ahmadinejad in 2005, although he has since reinforced them, not reduced their role in the economy.
Data on income distribution is murky. Certainly Ahmadinejad made some efforts to redistribute, including raising pensions and many salaries when he first took office. However, it appears that inequality is worsening. The gini coefficient may have been as low as .37 in 2003, although was probably higher. It appears to have fallen after 1979, but then increased again later above .40, possibly declining again in the early 1990s after the end of the Iran-Iraq war. The most recently reported figure I have seen comes from early 2008 from Abbas Bakhtiar (http://www.payvand.com/news/o8/jan/1258.html) at .44 and supposedly rising. Bakhtiar also reports that the sanctions by foreigners have had a negative effect on the Iranian economy, although he blames corruption and internal problems, including a stifling influence coming from the bonyads, as primarily responsible.