I am a US permanent resident and I have a Form I-131 travel document. I want to visit Ireland. Can I enter Ireland with these, or am I required to apply for a visa?
The entry requirements to Ireland are determined by your nationality. Neither I-327 (permit to reenter), nor I-571 (refugee travel document) grants you any specific rights with regards to being admissible to Ireland.
So, the answer to your question “Can I enter Ireland with these, or am I required to apply for a visa?” is, “That is determined by the national passport you carry.” If you do not carry a national passport, both I-571 and I-327 require a visa.
Your country of residence is mostly irrelevant for traveling to third countries; so is your Green Card: the US government gave you a document, why would you assume it has any meaning to the government of another, independent country, here Ireland?
What you need to enter Ireland depends on your nationality, which is shown by your passport. You didn't specify that, so there is no way to answer this.
Having a Green Card can help to positively influence a visa decision, because many countries consider the underlying decisions of the US a worthwhile input, but it doesn't bestow any rights outside the US.