You are looking for parish records. You can search an
Irish Heritage Foundation for birth and marriage records
depending on what records have survived. You have both names which is a real help in your search.www.rootsireland.ie
Griffith Valuation 1857: www.askaboutireland.ie
There are 432 Murray's of which 14 named Peter
and 44 Burns
The parish's in common are Drum, Killukin, Kiltullagh, Roscommon.
In 1822 there is a birth for an Eliza Burnes 26 Aug. 1822 fa: johanes (John) Burnes and mother Elizabetha Loughna, Boyle, Roscommon
Elizabeth 7 April 1830, Boyle
Patritii Burns and Bab Noon
These are from a CD from the Latter Day Saints - no longer
available - see www.familysearch.org
They are records from parish's whose records were not destroyed in the 1922 Four Courts Fire in Dublin
There is no marriage for this couple in the CD records I have but there are many more parish's.
Tithe Applotments 1823-1837: www.genealogy.nationalarchives.ie
212 Murray's 18 named Peter if Peter is named after father'
18 Burns, you could try comparing Parish's again to match up with the Griffith Murray & Burns.
The current on line telephone books shows 39 Murray's
and 4 Burns. Its hard to believe but you may find some families still in situ. You could try writing the 4 Burn's and see if you get any answers and you never know!
Would they still have been in Ireland for the 1901 or 1911