Yes, to the best of our knowledge you should be able to use your valid Texas driver's license to drive a rental car throughout Ireland (From Dublin up to Belfast in North Ireland).
Both Ireland, the United Kingdom (the UK encompasses England, Scotland, Wales and Northern Ireland), and the United States are members of the UN Convention that allows those with foreign licenses to drive in their country. Over 100 other countries are party to the International Convention on Road Traffic that we are referring to. This Convention is intended to establish uniform reciprocal privileges on matters such as drivers' permits. The agreement also requires countries to allow foreign drivers admitted to their country to drive a private motor vehicle for one year on certain conditions.
The US Embassy in Ireland states that American citizens are permitted to drive with a U.S. driver's license for the duration of a visit to Ireland; i.e., as long as their status is that of tourist and not resident. Once you decide to become resident in Ireland, regardless of how long you have been in the country, you must apply for an Irish driver's license.
The American Consulate also advises that driving is on the left side of the road in Ireland so motorists without experience in left-drive countries should be particularly cautious. Tourists driving on the wrong side of the road are the cause of several serious accidents each year. Road conditions are generally very good, but once off main highways, country roads quickly become narrow and uneven. Roads are more dangerous during the summer and on holiday weekends due to an increase in traffic.
According to the Citizens Information, an Irish eGovernment website, if you hold a national driving license or an international driving permit from your own country, you may drive in Ireland for the duration of your temporary visit (up to 12 months) normally. Since 1 January 2003, you are required to carry your driving license with you at all times when motoring in Ireland.
You may also want to obtain an international driving permit (IDP). An IDP to Ireland means a valid and property completed international driving permit issued to a visitor under the Convention of 1949 by a competent authority of the State/Country in which you normally reside. You cannot obtain an IDP outside of your country of residence, so you would need to obtain this in Texas before traveling to Ireland if you wanted to obtain this document to carry with your. You can get an IDP at AAA here in the United States.
The Driver and Vehicle Agency of Northern Ireland (DVLNI) notes that a visitor may drive vehicles up to 7.5 tons and with up to 16 passenger seats not for hire or reward, provided your full license or driving permit remains valid for up to 12 months from the date of entering NI.
If you have more questions about driving in Ireland and Northern Ireland read through the Citizens Information and contact this Irish government agency and/or the DVLNI if necessary. You may also want to contact the Irish rental car agency you plan to use to find out what all documentation they require you to show when picking up the vehicle you plan to drive for the length of your vacation. Hope you have an enjoyable and safe trip over to Ireland.