I have not yet found an RC burial record for Patrick Galligan/Gallaghan, who was born about 1807 in Co. Cavan (probably parish of Kilmore), Ireland, and who emigrated to Canada about 1843. I have checked a number of Roman Catholic parish registers (e.g., St. Michael’s, Corkery; St. Michael’s, Fitzroy Harbour; St. Peter Celestine, Pakenham), but so far, no burial record. It may be that I am overlooking something obvious; it may be that I am overlooking something obscure. Or perhaps his burial was recorded and the record was subsequently lost, misplaced, or destroyed. Or perhaps his burial was never recorded in a parish register at all.
In any case, despite the lack of a church burial record, I do have three different records of the death or burial of Patrick Galligan:
1. Patrick Galligan is listed in Schedule No. 2.–Nominal Return of the Deaths within last twelve months in the 1871 census of Canada (Ontario, Carleton, Fitzroy Township):

Note that the “Disease, or other cause of Death” (column 11) is given as “Disease of heart.” Patrick Galligan is listed as Sex: Male; Age: 63; Religion: R. [Roman] Catholic; Country or Province of Birth: Ireland; Profession, Occupation, or Trade: Farmer; Married or Widowed: W [Widowed]; Month of Death: April.
2. There is an Ontario civil death registration (County of Carleton) for a Patrick Gallaghan, Farmer, who died 25 April 1870 at 63 years of age, and who was born in Cavan, Ireland:

accessed 23 January 2013), Ontario, Canada, Deaths, 1869-1938 and Deaths Overseas, 1939-1947.
Note that the cause of death is here listed as “Disease of the heart 3 years,” which corresponds with the “Disease of heart” given in the 1871 census record of Patrick Galligan’s death. Note, also, that the death informant, his son Denis, did not sign but rather marked with an X. If you want to know whether or not your ancestors were literate, look for records which include a signature, and then look for either an actual signature, or else an X with “his mark” or “her mark.”
3. There is a headstone (for “Patrick Gallaghan, died April 25, 1870, A.E. 63 Yrs.”; buried with “[M]ary Cullen, His Wife”) at St. Michael’s RC Church and Cemetery, Corkery, Huntley township (Carleton Co., Ontario). Photo of said headstone posted at the Canadian Gravemarker Gallery for Ontario/Eastern Ontario/Ottawa-West of the Rideau/Huntley Township/St. Michael’s RC.
For 19th-century Irish emigrants, deaths/burials are easier to find than births/baptisms. Which is to say, a record of a death or burial (in Canada, the U.S., Australia, or etc.) is more likely to be found/is more likely to exist than a record of a birth or baptism (in Ireland). And the possibility of a headstone has to be entered into the Likelihood of a Record column, under Deaths/Burials. Unless your ancestor was absolute true royalty, it’s unlikely that his or her birth was marked by a lasting record, with an official portrait and a name memorialized in ‘black as ink’ marble stone. But when it comes to death and burial in the New World: well, even a humble farmer from County Cavan, Ireland who emigrated to Upper Canada in the mid-1840s (like, for example, my 3x great-grandfather Patrick Galligan) might have his name and his date of death carved out in rough stone, for posterity.