Shelbourne will host the Premier League's Crystal Palace in the league phase of the Conference League.
Joey O'Brien's side's other home games will be against Kosovo's Drita, and Hacken of Sweden.
While their away fixtures will see the Reds face Shkendijan of North Macedonia, Slovenia's Celje and Dutch club AZ Alkmaar.
Shamrock Rovers matches in Tallaght will be against Shakhtar Donetsk, Celje and Malta's Hamrun Spartans.
The Hoops will travel to Sparta Prague, AEK Athens of Greece and Iceland's Breidablik.
In the Europa League draw, Nottingham Forest will host manager Nuno's former club Porto when they return to European action in the Europa League.
They will also meet another Portuguese side, Braga, and have a trip to last season's Conference League runners-up Real Betis.
Aston Villa go up against Turkish team Fenerbahce - who sacked manager Jose Mourinho earlier on Friday.
After missing out on Champions League qualification, Celtic have matches against Roma and Feyenoord.
Rangers also have a home game with Roma and take on Porto and Braga.