Iran has been considered an "outcast" nation for many years. Iran's new Prime Minister, Hassan Rouhani, has been making excellent strides to correct this situation. At this point in time, Iraq's military forces appear to be ineffective. Calling upon Shiite militia to take a stand against ISIS is quite likely to lead to revenge killings of Sunnis, given the recent reported atrocities of ISIS. Enlisting the services of professional military could help. In return, this will provide the opportunity for Iran to be viewed as a productive contributor to a dangerous situation, which should help its ongoing search for greater acceptance and legitimacy.
Yes, Iran will benefit from sending government forces to assist, because Iran is not a friend of the United States or of Iraq. Iran sees a weakness in Iraq. Iran wants to exploit the opportunity to come in and take over. Iran will have an increase in power in the region if they send government forces to assist.
There will always be a financial benefit from coming to the 'aid' of others and Iran seems like it can weather the storm that comes from falling in on either side of what's going on in Iraq. Whatever the outcome, I'm sure that Iran can use the situation to their advantage.
First of all Iraq hasn't officially asked Iran for help and I don't believe they should just send troops to assist where they aren't wanted. Second of all I'm not sure the United States wants Iraq to be even more influenced by Iran. I believe the best course is for them to stay out of it.