The financial argument is that, after over twenty years of being a net beneficiary from Europe, if the EU expands to take in poorer eastern European countries then Ireland would become a net contributor. But this is an over-simplification - it doesn't tell us who actually receives the money that the EU sends to Ireland. Its not like we each get an equal share of any EU money that goes to Ireland, in fact a lot of it ends up in the hands of those who need it least, Larry Goodman and others like him.
The National Platform also argue that if the Nice Treaty is passed it will see thousands of eastern Europeans move west in search of work. (Actually, Germany and Austria are arguing that eastern Europeans should be second-class citizens for a few years after their countries join the EU.) This is the complete opposite of one of the main anarchist arguments against the European Union - that it has allowed the free movement of money, but actually prevents people from outside Europe from moving here in search of jobs. We want a Europe, and a world, without any borders, and no restrictions on freedom of movement.
The main argument proposed by the National Platform is that the reduction in voting rights and extension of qualified majority voting agreed in the Nice Treaty amount to a reduction in 'national sovereignty'. But this is nothing that anarchists should care about because, from our perspective, it doesn't make much sense. The Irish government doesn't represent our needs, and it doesn't act according to our wishes. It passes laws and budgets that are in the interests of the ruling class, not of the ordinary Irish worker. So its not 'our' power that's being lost if the Treaty passes, because we don't have that power anyway - we'll just be exchanging one set of rulers for another, slightly different set of rulers. There are plenty of good reasons to vote against the Nice Treaty, but this isn't one of them.