From the result of inquiries, which the Directors have caused to be made, into the system of Scotch railways, it is the intention of the Board to use animal power exclusively until the line is finished half way to Drogheda; and even when finished to Drogheda, it is intended that all the goods will be carried by horse power, and in the intermediate hours, when no steam train shall start for Drogheda; that a horse train shall run from Dublin to Malahide, and another from Malahide to Dublin, as this division of the line must be supplied with extra means for its own peculiar traffic, which will not be required on the rest of the line.
Report to the Proprietors of the Dublin and Drogheda Railway in The Railway Magazine; and Annals of Science No XXVI April 1838
Irish waterways history 
Not that it transpired that way. There is no record that the D&DR used horse power when it opened. Excluding the city tramway networks, the use of horse power for service trains on railways in Ireland was limited. A few that I am aware of are:
the temporary working of the Cork & Youghal Railway’s initial section from Bruin Lodge (Cork) to Tivoli;
a section of what became the Cork, Bandon & South Coast Railway;
the very short lived Magilligan to Magilligan Point line (ran for 4 months in 1855!);
the Fintona Horse Tram in Co. Tyrone, which remained horse worked for passengers until closure in 1957.