There's no problem that you apply for a residence card in Ireland and your application won't be turned down purely because you have been refused by UK before. However, you should be careful the reason, for which your UK application got refused, isn't there any more. There must be some quite solid ground if they refused you for 3 times.
Surinder Singh Settled 2019
Form received by HO: 30th Sep
Passport returned and asking for a more recent photo: 11th Oct
New photo received by Home Office: 14th Oct
Biometrics letter received: 23th Oct
Biometrics enrolled: 26th Oct