Amid heightened tensions following the 19 June downing of a US Navy unmanned aircraft system by the Iranian military over the Persian Gulf, the US Federal Aviation Administration has banned all US carriers from operating within the Tehran FIR over the waters of the Persian Gulf and Gulf of Oman (highlighted area pictured below).

Notam text
The FAA issued the following Notam at 01:48 UTC on 21 June.
A0019/19 (Issued for KICZ PART 1 OF 2) - SECURITY..UNITED STATES OF AMERICA PROHIBITION AGAINST CERTAIN FLIGHTS IN THE OVERWATER AREA OF THE TEHRAN FLIGHT INFORMATION REGION (FIR) (OIIX) ABOVE THE PERSIAN GULF AND GULF OF OMAN ONLY. ALL FLIGHT OPERATIONS IN THE OVERWATER AREA OF THE TEHRAN FLIGHT INFORMATION REGION (FIR) (OIIX) ABOVE THE PERSIAN GULF AND GULF OF OMAN ONLY ARE PROHIBITED UNTIL FURTHER NOTICE DUE TO HEIGHTENED MILITARY ACTIVITIES AND INCREASED POLITICAL TENSIONS IN THE REGION, WHICH PRESENT AN INADVERTENT RISK TO U.S. CIVIL AVIATION OPERATIONS AND POTENTIAL FOR MISCALCULATION OR MIS-IDENTIFICATION. THE RISK TO U.S. CIVIL AVIATION IS DEMONSTRATED BY THE IRANIAN SURFACE-TO-AIR MISSILE SHOOT DOWN OF A U.S. UNMANNED AIRCRAFT SYSTEM ON 19 JUNE 2019 WHILE IT WAS OPERATING IN THE VICINITY OF CIVIL AIR ROUTES ABOVE THE GULF OF OMAN. A. APPLICABILITY. THIS NOTAM APPLIES TO: ALL U.S. AIR CARRIERS AND COMMERCIAL OPERATORS; ALL PERSONS EXERCISING THE PRIVILEGES OF AN AIRMAN CERTIFICATE ISSUED BY THE FAA, EXCEPT SUCH PERSONS OPERATING U.S.-REGISTERED AIRCRAFT FOR A FOREIGN AIR CARRIER; AND ALL OPERATORS OF AIRCRAFT REGISTERED IN THE UNITED STATES, EXCEPT WHERE END PART 1 OF 2. 21 JUN 01:48 2019 UNTIL PERM. CREATED: 21 JUN 02:00 2019
A0019/19 (Issued for KICZ PART 2 OF 2) - THE OPERATOR OF SUCH AIRCRAFT IS A FOREIGN AIR CARRIER. B. PERMITTED OPERATIONS. THIS NOTAM DOES NOT PROHIBIT PERSONS DESCRIBED IN PARAGRAPH A (APPLICABILITY) FROM CONDUCTING FLIGHT OPERATIONS IN THE ABOVE NAMED AREA WHEN SUCH OPERATIONS ARE AUTHORIZED EITHER BY ANOTHER AGENCY OF THE UNITED STATES GOVERNMENT WITH THE APPROVAL OF THE FAA OR BY A DEVIATION, EXEMPTION, OR OTHER AUTHORIZATION ISSUED BY THE FAA ADMINISTRATOR. OPERATORS MUST CALL THE FAA WASHINGTON OPERATIONS CENTER AT 202-267-3333 TO INITIATE COORDINATION FOR FAA AUTHORIZATION TO CONDUCT OPERATIONS. C. EMERGENCY SITUATIONS. IN AN EMERGENCY THAT REQUIRES IMMEDIATE DECISION AND ACTION FOR THE SAFETY OF THE FLIGHT, THE PILOT IN COMMAND OF AN AIRCRAFT MAY DEVIATE FROM THIS NOTAM TO THE EXTENT REQUIRED BY THAT EMERGENCY. THIS NOTAM IS AN EMERGENCY ORDER ISSUED UNDER 49 USC 40113(A) AND 46105(C). ADDITIONAL INFORMATION IS PROVIDED AT: HTTPS://WWW.FAA.GOV/AIR_TRAFFIC/PUBLICATIONS/US_RESTRICTIONS/. SFC - UNL END PART 2 OF 2, 21 JUN 01:48 2019 UNTIL PERM. CREATED: 21 JUN 02:00 2019
Practical effects
Very few US carriers operate flights through the Tehran FIR and less than a handful of those cross the now prohibited areas. Flights that do are generally cargo operators. Earlier on June 20 United Airlines canceled its service returning to New York (EWR) from Mumbai and has canceled flights on the route until further notice. While the flight does not usually pass over the prohibited area, United said a thorough safety and security review prompted its actions. UA48 and UA49 both regularly transit overland portions of the Tehran FIR (see below).

The real effects may felt if carriers in other countries decide to follow the FAA prohibition or if other national authorities issue their own prohibitions.
Background on Iranian airspace
In 2014, security concerns above Iraq prompted much of the traffic between the Gulf states and Europe and North America to begin transiting Iran. In November 2017, Iraqi airspace reopened to transiting traffic, pulling in traffic that had previously transited Iran. The below images show the regional airspace in on 21 June from 2017, 2018, and 2019.
There are a few changes worth noting. From 2017 to 2018, the concentration of transiting flights to and from Europe/North America between the big 3 hubs—Dubai, Doha, Abu Dhabi—shifts from Iran to Iraq. From 2018 to 2019 flights that previously transited Pakistan, Afghanistan, and Turkmenistan now primarily route through Iran due to the continued airspace restrictions put in place by Pakistan. This has also pushed much of the transiting traffic south, nearer to the area now prohibited to US carriers.
This is a developing story and updates will be posted in this post as necessary.