I think there are 5 or 6 about the right age, on the 1861 transcriptions on Ancestry and from memory 3 are married. In many cases these are pretty awful transcriptions, so could easily be mistranscribed and missed.
Thing is James may not necessarily have stayed in the area, may have married or died, immigrated, moved to England, or back to Ireland .... anything ... I suppose all you have to go on is place of birth place and age - you don't even know his occupation.
Have you tried searching Scotland's People for a death for him (using his MMN if you know it) but not specifying area? I entered James Burns b 1838 +/- 5 years, searching 1855 to 1891 and got 41 matches over 2 pages, so not too many. You might need to search 1851 to 1855 on the OPR's as well as extending the end year. It's really all I can think to suggest ... sorry.
Although you can't find him in 1861 have you searched for him in 1871 and later? Sometimes the odd person can't be found in a census or two.
