Can an Iranian citizen attain ATP in the United States?
The U.S. does not prohibit Iranian citizens entering the United States and remaining here on extended business. It also allows noncitizens to attend flight school, conditional on a review of the candidate by the Transportation Security Administration. So, the U.S. would not categorically prohibit an Iran citizen learning to fly and obtaining a commercial or airline pilot's license.

However, while not impossible, it will be much harder than I just made it sound if you are living in Iran right now, because communications between Iranians and the U.S. State Department have been made difficult by the politics of the past 35 years.

There is, to put it delicately, no functional United States Embassy on Iranian soil. The U.S. maintains a virtual Iranian Embassy online at http://iran.usembassy.gov/ (also in Farsi at http://persian.iran.usembassy.gov/); however it was blocked by Iranian information authorities less than 12 hours after it initially opened and remains blacklisted to this day. Users with a VPN tunnel outside Iran's firewall can access it, but the site is mostly educational; you cannot apply for a visa using the site alone. You'd have to be able to submit your paperwork to a functioning U.S. Embassy or directly to the State Department offices in Washington D.C., which will be a delicate process as the Revolutionary Guard will be monitoring both sides of such correspondence.

Assuming you can establish reliable contact with the U.S. State Department either directly or through an Embassy, you would need an extended visa. There are three basic reasons to stay longer than 90 days, with visa classes for each; work, school, and temporary asylum. If you are coming to the U.S. specifically to learn to fly, an M-1 visa should be what you apply for; it permits entry to the U.S. for "vocational or other non-academic training", including FAA Part 61 and Part 141 flight schools. The Iranian authorities also look most favorably on student visas compared to other types.

In addition to the visa, you would, like any other foreign applicant for flight training, be required to pass a TSA security clearance. This is ultimately discretionary; you cannot force the TSA to grant your clearance (ask any U.S. national; you can't force the TSA to do anything). 

While neither the State Department, TSA nor the FAA would categorically prevent you learning to fly based on country of origin, the TSA does list Iran as a country of origin that prohibits the visa holder from "Automatic Revalidation". Essentially, if while attending flight school, you ever left the U.S. (to visit family back home, or even driving north to Canada or south to Mexico for sightseeing) you would be required to go through the full security clearance process again including the TSA check. So, you would have to stay in the United States political borders at all times while in flight school, or you could risk being denied reentry or resumption of your flight training.

While taking flight training and amassing your 1000 hours flight time required for an Airline Transport Pilot license (that 1000 hours number is good only if you're attending a four-year flight school; for most civilians working their way up it's 1,500), there are a number of illegal acts you can unintentionally commit while flying. For citizens, the legal penalty for unintentionally breaking an FAA regulation ranges from nothing to a talking to by air traffic control up to formal hearings with possible suspension or revocation of your pilot certificate. For a non-citizen in the United States for the express purpose of learning to fly, any offense bad enough to get your pilot's license suspended would likely also get you deported (if not for the criminal act, then because you can no longer do what you're here to do). But, you'd usually have to mess up pretty bad several times to get that far.

Once you have your ATPL, your M-1 would no longer be valid as you'd no longer be a student, and so you'd have to leave the U.S. unless you made prior arrangements with the State Department and/or an airline to remain in the country under another status (such as H-1b to work for an airline). If you take your license and leave the country, the FAA ATPL would only allow you to fly "N-numbered" aircraft registered by the FAA. Most other countries have their own registration, and would require you to "convert" your pilot's license to the local regulatory agency's version before you could fly a locally-registered plane. For an ATPL holder this is typically easier than for lower levels, and for the EASA (the largest ICAO-compliant regulatory body besides the FAA) it boils down to a skills test (knowledge and practical), a new medical cert, and being re-rated by the EASA for multi-engine operations (the FAA AMEL class rating does not transfer).