A friend is a citizen of an EU Schengen country and is resident in that country. His wife is a Chinese (PRC) citizen. She has permanent residence but not citizenship. They would like to visit the Republic of Ireland which is EU but not Schengen. Does she need a visa?
A visa (free of charge) is likely required.
Family members of EU/EEA/Swiss citizens exercising their right of free movement do not need a visa to enter Ireland, only if the family members have a residence card granted under EU law specifying that they are the family member of a union citizen or referring to Article 10* of EU Directive 2004/38/EC.
A friend is a citizen of an EU Schengen country and is resident in that country.
However, one cannot exercise the right of free movement within one's own country** since one did not "move" and the right to live in one's own country is implied by one's national citizenship instead of EU citizenship. EU law does not regulate reunification of family members of a member state's own citizen in the member state (e.g. the spouse of a German citizen living in Germany can only immigrate to Germany under German law).
So the wife of your friend is likely to only hold a residence permit under national legislation and not EU law. In this case, she cannot benefit from the visa-exemption based on EU family member status.
In any case, if a visa is required, Irish short-stay visa is free for
- family members (regardless of citizenship) of an EU citizen (regardless where they live), if the family members are travelling with the EU citizen, or if the EU citizen is already in Ireland (proof must be provided); or
- certain non-EU nationals (only citizens of a country eligible for the a special Irish program*** qualify; China is included) holding a long term residence card (no matter which kind) in a Schengen country.
Either fee exemption could apply to a Chinese national, but the second basis is probably more straightforward.
New residence cards issued to family members of EU citizens under EU law should in principle be uniform in that they should have "(permanent) residence card" in the title and read "Family Member EU Art 10 (or 20) DIR 2004/38/EC" on the front of the card. Residence permits issued under domestic law for family reunification usually have the title "residence permit" instead and do not refer to EU legislation.
But there are still other valid residence cards (in various format, e.g. paper-based) in circulation issued before they were made uniform; and sometimes countries do not follow the regulations exactly. They are still valid for Irish entry as long as it bears the name or inscription "Residence card of a family member of a Union citizen", or otherwise makes clear that the holder is a family member of an EU citizen and the permit is issued under EU law.
* Literally read, only article 10 cards qualify for visa exemption pursuant to Article 5(2) of Directive 2004/38 and according to Irish immigration. However, ECJ ruled in 2020 that the article of the EU directive must be interpreted as such that article 20 cards also benefit from visa exemption, even if Irish (and other) immigration authorities may not have updated their websites and guidelines. You do risk being denied boarding by certain over-zealous or over-cautious staffs however, if you only have an article 20 card and their system does not recognize article 20 cards as visa-exempt. Additionally, if the family member is not joining or accompanying the EU citizen in Ireland, there may be questions on the applicability of the Directive. With article 10 cards, Irish immigration does not require the family member to be joining or accompanying the EU citizen.
** There are exceptions for EU citizens moving back to their own country after having exercised their right to free movement in another EU state with their family members. But not all countries have straightforward procedures in this case. The determinative feature is the inscription on the residence card.
*** I am not naming the program directly in that sentence because the name is misleading in this context. It is called Short stay visa waiver programme but the program is only applicable to certain situations where the traveller has first legally entered the UK. However, the list of countries eligible for this program is used to determine eligibility for the fee exemption even when a visa is required.