Northern Ireland fans have been angered by the travel restrictions placed on them for the games in Dublin.
Fans have been told that they must travel on official buses which will cost £30 for a return trip.
"We are calling on our members not to attend the games," said NI supporters chief Gary McAllister.
"It is in effect of boycott of the games against the Republic of Ireland and Wales.
"We feel that the restrictions that have been placed on us have been very badly handled."
The travel restrictions have been placed on the Northern Ireland supporters following minor incidents at the Nations Cup opener against Scotland at the Aviva Stadium in February.
McAllister told that BBC Sport that the Irish FA had ignored advice from the supporters group in advance of the February game.
"Our organisation advised the IFA to put restrictions on the sale of tickets and transport for the game against Scotland.
"There were a few minor incidents at that game which have led to more drastic restrictions.
"If the IFA and the authorities had got it right first time around, they wouldn't have had to put in such severe restrictions for these games."
McAllister added that his organisation are unhappy at the "travel monopoly" that has been created.
"If fans had been allowed to travel independently on their own buses, they would have been able to organise transport for themselves for around half the price they are being asked to pay."
Northern Ireland face the Republic on 24 May before taking on Wales three days later.