My partner and I (British nationals) are travelling to the Republic of Ireland with our son, who was born in Canada and holds a Canadian passport. Will this cause any issues when we board a plane? We are planning on also taking his birth certificate to prove we are his parents.
No, it's very unlikely to cause problems. There are three different nationalities in my family, we've traveled extensively with various permutations of them including the kids having different passports from the parents, and nobody at an airline or Immigration has ever even asked a single question about it.
Caveat: things may be more complicated if there's any reason for Immigration to suspect that you're not the child's parents and/or there are potential signs of child trafficking (see under "Trafficking Indicators"). But if the three of you are traveling together, the child shares your last name and looks related etc, this is very unlikely.