I've chosen to post an answer as this is a corner case not clearly explained by either the linked duplicate or the UK web site.
Is your residence permit the new format (IRP) which is red and blue and has a bull and stars above your photo? If yes, then you can transit without visa, as this is the EU common format residence permit, and the UK allows EU residents with such a card to transit without a visa. The old GNIB residence card is not valid for transit without visa through the UK.
This is an EU common format residence card issued by Ireland, also known as IRP (Irish Residence Permit):
You can also transit the UK without visa if you still have a valid type D visa from Ireland.
The relevant exceptions as quoted from the UK web site are:
You don’t need a [transit] visa if you have one of the following:
- a common format residence permit issued by an European Economic Area (EEA) country or Switzerland
- a uniform format category D visa for entry into a country in the European Economic Area (EEA) or Switzerland