The holders have been pitted against Drogs in the competition's last 16, with St Patrick's Athletic drawn against Dundalk in another all-Premier Division tie
Last season's runners-up St Patrick's Athletic host Dundalk, with the winners advancing to the quarter finals, while Bohemians drawn against Bray Wanderers in the other all-Airtricity Premier Division tie.
Leinster Senior League club Crumlin United face Swords Celtic after overcoming Bluebell United in the Third Round, while Sheriff YC travelling to Kilbarrack United in their last 16 tie. Wexford's Shamrock Rovers have been drawn against Longford Town after entering the competition in the First Round.
All the Fourth Round ties are scheduled to take place before February 16, 2014.
Leinster Senior Club Fourth Round draw:
Wexford Youths v Athlone Town
Kilbarrack United v Sheriff YC
UCD v Shelbourne
Shamrock Rovers v Drogheda United
Shamrock Rovers (Wexford) v Longford Town
Bohemians v Bray Wanderers
Crumlin United v Swords Celtic
St Patrick's Athletic v Dundalk FC.