I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. B4 1840 rcds are sparse, B4 1800 rare. My search is limited to this database.
For Daniel Burns B 1845 +/- 4 yrs in Co. Armagh, I found 0 B rcds. Extending the search to all Ireland, I found 17 B rcds; none had a father named Joseph.
For Daniel Burns M 1865 +/- 2 yrs in Armagh, I found 2 M rcds one civil the other church; these may be duplicates. A view of the detailed rcd SHOULD show the bride's name. If the name is Catherine, there is a good chance the rcd if for your ancestor. The marriage rcd SHOULD also show their parents' names, etc.
Google the Foundation's name to lead you to this database. Pls post your search results to my genforum msg.