Ireland, like every other country, has a general rule of no entry without (a) a passport (valid and not expiring soon), and (b)(i) a visa or (ii) a visa exception.
For example, to go to China, I need a valid passport that will not expire in the next 6 months. I am not aware of any visa exceptions for US citizens who want to go to China, so as far as I know, you need to mail your passport to a Chinese embassy/consulate, and have them stick a visa in your passport. They will most assuredly put your name in a database, and run a check on you. Those visas used to be only short term, requiring constant new visas to be obtained. But, as China has become more reliant on international trade, they started issuing visas which allow short stays, but limitless re-entry during a 10 year period. You have to leave and come back; you can't stay for 10 years. If you have no passport, or if you have a passport without a visa, then the PRC will deny entry. If you stay too long, you will be imprisoned/deported. Obviously, this is not open borders.
Every country is this way by default, unless the country unilaterally passes visa exceptions (or negotiates them with other nations). Common visa exemptions are by citizenship. For example, the UK might grant a blanket visa exemption to any US citizen with a valid passport, to come for a term (e.g., 30 days) for a specific purpose (e.g., business, or tourism). If you are not going for a specifically exempted purpose, you need a visa. If you plan on staying beyond the exemption term, you need a visa. Visas are granted at the discretion of the country you want to visit (within their discretionary statutory guidelines). Just because you traveled to a country like Ireland who granted a visa exemption to US citizens for short visits does not mean that Ireland has open borders. You can easily be deported by them by violating your visa exemption rules...like, if you stay a year and get a job or claim government benefits.
I am fine with this for Mexico, or any other Latin American country. The Democrat candidates are asking for much more, though, don't you agree?