I searched the Irish Family History Foundation's database. It contains all surviving rcds for the majority of Irish counties. ALL rcds B4 1840 are increasingly sparse. Rcds for Co. Dublin are in this database, but not the rcds for the City of Dublin.
For William Byrne, I found several B rcds, but none in 1811 (exact). There are 3 possibilities: 1) the B event wasn't rcd'd, 2) the rcd didn't survive, or 3) the data hasn't been digitized.
You can google the Foundation's name, register, opt to be notified as rcds are added, and browse. Or, you can try to contact church authorities in the city of Dublin and try to find his B rcd there.