Upvote:1
From the tone of your message you seem to be a US citizen. US citizens don’t need a visa for either the UK or Ireland. You are processed on arrival and as long as you can prove you’re able to support yourself and intend to return you will simply be granted a tourist leave stamp in your passport. This will be given to you when you enter Ireland through an official port of entry.
If you are not a US citizen and require a visa for entry to the UK you will also require a visa for entry to the Republic of Ireland. Immigration laws are harmonised between the two countries to preserve open borders between Northern Ireland and the Republic but they still operate separate regimes and don’t accept each other’s visas for entry. Check out citizensinformation.ie for info on your specific circumstance.
Upvote:3
The British-Irish visa scheme only applies to visas that are issued through a joint visa application centre in China or India. It is therefore not available to you if you need to apply in the USA.
Your itinerary will therefore require you to apply for separate visas for the UK and Ireland.
As there's an open land border between Northern Ireland and the Republic of Ireland, it is possible that you can reach Dublin without being stopped by anyone even if you don't have a visa for Ireland. In that case you would almost certainly be allowed to leave the Republic by air. However, authorities will be able to deduce from airline passenger information that you must have entered Ireland without a visa, and this is likely to create problems in any future visa application to Ireland, the UK, or anywhere else they might share data with.