Sorry you have been waiting a long time for a reply! I do not check some of these sites v often. I have been doing lots of work on fh stuff though.
I am not sure what progress you have made. But if you check Peter maddens wonderful site with lists of tippereary born people who came to Australia, you will see Dennis burns arriving on the Larne in 1841. He is aged 35, son of James and nancy, from Tipp/Tipp which I think means Tipperary town, both parents dead, with Margaret, 34, d of John and Margaret Banbury, also of Tipp/Tipp, with children Anne, 13, Margaret 11, Edmond, 7, and Michael, 1.
Hope after all that waiting that is news to you and helpful. Not sure what connection there is with my byrnes snd prob not possible to find out either. If you're still working on it I can search a bit further for you. There is the new National Archives of Ireland site which now has a search facility of Tithes documents for around 1830. You may find ref to Dennis Byrne/Byrnes/burns or his father, or the Banbury's there.
Best wishes. Feel free to email me directly on firstname.lastname@example.org