This is a very complicated situation, and I think we do not have full information about all that is going on. However, while some of the protests have aimed at Rohani, increasingly some of it has been directed at the top leader, the Vali-e-faqi, or Supreme Jurisprudent, the unelected Ayatollah Ali Khameini, with reports of crowds chanting “Death to Khameini” and burning photos of his face. It also should be noted that while not nearly as deadly as the Green Movement demonstrations against apparent electoral fraud in 2009, they seem much more widespread across many cities in Iran, while the 2009 events were largely in Tehran and a few other largest cities.
It is important to keep in mind how power is held and distributed in Iran as one sees all kinds of characterizations about it, including declarations that Iran is a “dictatorship.” It is not, but it is true that the unelected leader (Khameini) has more power than an elected one (Rouhani). In particular, Khameini is the Commander-in-Chief of the military as well as being in charge of the judicial system based on Shia Sharia, as the proper translation of his official title as “Supreme Jurisprudent” indicates. While he does not directly control them, it is the clerical hierarchy under him, along with parts of the military, that control the bonyads that constitute probably more than a quarter of the economy, which also has indicative planning and a substantial state-owned sector. This latter part is more under the control of the elected president and his economics minister, as well as having more control over the Iranian central bank.
The “Poland problem” part involves those parts of the economy that can be influenced by Rouhani and his secular ministers and bureaucrats. Somewhat like in Poland, he can be partly blamed for not increasing redistribution or aid for the broader population. The part he does not control is the massive corruption tied to the bonyads and the clerically controlled parts of the economy. Long simmering unhappiness over this corruption appears to have finally exploded, although Rouhani and his government are also being blamed. I have no idea where this is going, but I fear that many more could end up dead than the two who have been killed so far reportedly.
I must make a comment about the incoherent response by President Trump. It looked to be true that a target of the protests has been increasing funding going to the military for Iran’s foreign adventures. But Trump supports the same thing in the US, even though he ran against such a policy. He is also moving to increase both inequality as well as “swampy” corruption, even though he also ran against those.
Of course the biggest problem from him has been his ongoing efforts to undo the JCPOA nuclear deal that led to what economic improvement Iran has experienced. He has resisted removing remaining sanctions (in place against human rights violations and missile development), thus aggravating the Iran economic problems, and he clearly wants to simply end the JCPOA and reimpose harsher sanctions. His proclaimed sympathy for protesting Iranian citizens looks hypocritical (along with the fact that he is blocking any Iranians wanting to escape to the US from doing so).
A final point many do not realize. Those who want to end the JCPOA have also called for a regime change end of the theocratic regime, which could happen. However, those pushing for this, many of whom thought the US should have given more active support to the failed Green Movement in 2009, do not realize that civilian nuclear power is very popular in Iran. The Green Movement supported civilian nuclear power, even as they did not support it military nuclear program. This was basically the position of Rouhani, who was not only elected on such a platform but reelected, even as Trump has ignored that and praised the truly dictatorial absolute monarchy of Saudi Arabia. If indeed the theocracy of Khameini were to be overthrown, Iran would almost certainly continue to pursue a civilian nuclear power program, if not a military one.