1) I just wanted to cover this in slightly more detail. The key thing to note with Iran's government is it actually has two distinct halves. A democratically elected side, and a religious side.
2) Here's a nice flowchart that helps sort it out. Now, a few things to note that makes the 'distinct halves' bit fuzzy. 
3) First, the Guardian Council, which is put together by the Supreme Leader and the Judiciary, vets ALL the candidates on the elected side. This includes the Assembly of Experts. Nobody gets into the government the religious side doesn't explicitly allow.
4) Second, the Assembly of Experts SOLE JOB is to manage the Supreme Leader position. This sounds more powerful than it does; it has turned completely vestigial, as they aren't actually allowed to oversee anything and cannot remove the Supreme Leader from power.
5) Third, the Expediency Council is meant to resolve disagreements between the religious and the non-religious halves of the government. However, it is totally under the control of and appointed by the religious side of the government. So guess how THAT usually goes.
6) Fourth, though more leading off of the previous one, is that the cabinet for the president has over time been submitted further and further to the approval of the religious half of the government.
7) For example, a few years ago, the Expediency Council ruled that the head of the Central Bank, normally appointed by the President, had to be run through the Guardian Council now.
8) That worked out great for the Mullahs. Now they had control of Iran's central bank, so they could get it to print an endless supply of Rials for the Quds Force to launder through UAE monetary exchanges for a currency worth a damn.
9) At this point, there's only 2 core structural problems with Iran's government that simply chopping away all the religious bits won't solve.
10) First: The judiciary is under the control of the religious side. Thus the religious laws being predominant in Iranian culture. The Iranian people will need to figure out how to get that back under control.
11) Secondly, the military, as in the actual military and not the paramilitary forces of the IRGC and Quds Force, are under the control of the Supreme Leader as well. So that will have to be resolved.
12) Otherwise, pretty much the entirety of the religious side of the government is pretty much vestigial. Remove the supreme leader, the guardian council, the expediency council, and the assembly of experts, and its like Wictor said, Iran has a government that can function.
13) Its corrupt as hell and will probably need to go through some DRASTIC personnel changes, but that's socialism for you. /end
