Unfortunately still nothing promising showing up - and no baptisms for any of the children so far .... I suspect this family were non-Catholic, possibly Church of Ireland or Presbyterian ?
The complication if the family are non-Catholic is that mother's maiden surname is not usually shown on baptisms, which will make them much harder to spot, and 'Dublin' can refer to the City, suburbs or County so covers quite a few parishes. Even though many church records have been made available on line over the last few years e.g. IrishGenealogy (free), the Catholic Records on NLI (also free) and searchableCatholic transcripts on Ancestry & FMP, not as many for Church of Ireland parishes outside the city and very few for other denominations on RootsIreland.
What denomination of Church did Arthur and his siblings marry in ?
Burns can be a variant of Byrne, which is a very common surname, but the two surnames that stand out in this case are McCluskey and the rare Oswald. There are a few Oswald records mentioned in the Dublin City records, but nothing that I could locate for a possible marriage or baptism for Ellen.
The surnames sound to me like there may be an Ulster origin and both McCluskey and Oswald are more likely to be found in Ulster counties in the 1850s, Co. Derry/Londonderry for McCluskey and Co. Down for Oswald, and similarly for Burns, if you only search for that exact spelling, with most in Co. Down followed by counties Armagh & Antrim.
I've had a scan though several 1850s Thom's directories to check for any John Burn's as a Cabinet maker or similar, the only close match in 1852 is for a John J. Byrne, cabinet maker at 6 Henry St., Dublin city. Based on this 1853 baptism this John Byrne was Catholic and married to Theresa. Directories usually only show the principle name for a business, so it's possible that your John Burns worked for one of the other Cabinetmaker businesses in the city.
