Traveling with Firearms

Any time a person departs the United States by any means or method of transportation, and is traveling with firearms and/or ammunition in their possession, the person must comply with all applicable laws and regulations governing the lawful exportation of these controlled items.
All persons who intend to travel from the United States to a foreign country with firearms and/or ammunition are reminded that both the permanent and temporary exportation of these items are subject to federal export licensing regulations.  The export regulations for handguns, rifles, associated parts and components, and related ammunition are found in the International Traffic in Arms Regulations (ITAR) administered by the Department of State’s Directorate of Defense Trade Controls (DDTC).  The export regulations for sporting shotguns (barrel length of 18 inches or more), muzzle loading firearms, associated parts and components, and related ammunition are found in the Export Administration Regulations (EAR) administered by the Department of Commerce’s Bureau of Industry and Security (BIS).
Export regulations require that prior to traveling outside the United States with firearms and/or ammunition all departing persons must obtain a valid and appropriate DDTC or BIS export license or qualify for a valid DDTC license exemption under 22 C.F.R. 123.17 - 123.18 or a valid BIS license exception under 15 C.F.R. 740.14(e).  
Before exporting any firearms and/or ammunition with a valid DDTC or BIS export license or a qualifying license exemption, the traveler, or an agent acting on the traveler’s behalf, must file the Electronic Export Information (EEI) using the Automated Export System (AES) or the Internet-based system AESDirectwhich is publicly availableand free of charge. In addition to filing the EEI in AES or AESDirect prior to export, all firearms, ammunition and additional mandatory documentation (e.g., certifications, foreign import permits, proof of AES filing; such as the Internal Transaction Number) must be presented to U.S. Customs and Border Protection (CBP) authorities for visual inspection at the port of departure from the United States.
Homeland Security Investigations (HSI) is the primary federal law enforcement agency responsible for investigating international smuggling operations and enforcing U.S. export control laws. Failure to comply with the federal regulations governing the temporary and permanent export of firearms and/or ammunition from the United States (including the proper filing of EEI) may result in the detention, seizure, and forfeiture of improperly declared firearms and ammunition and could further subject the traveler to arrest and criminal prosecution by HSI special agents for violation of federal export and/or arms smuggling laws.

https://www.ice.gov/cpi/quiz

https://help.cbp.gov/app/answers/detail/a_id/323/~/traveling-outside-of-the-u.s.---temporarily-taking-a-firearm,-rifle,-gun

Ireland and Italy both allow firearms with proper registration.  From everything I have read you must apply for an export license.