If an Irish business receives payments from foreign companies, do the Irish company have to pay tax on these payments. Assume that these payments are regular if that matters.
Ah come on now, how's anyone supposed to answer that?!
Has the Irish business performed work, provided a good or service, has it invoiced the foreign company - and if yes to any of the above, what was the nature of the work - in other words if you explain what's actually going on you might get an answer!
It's not necessarily as straightforward as that - the key determinant for VAT is the place of supply, and there are many rules (with various exceptions to the rules) depending on what type of supply it is, i.e. goods or services, and then specific types of goods or services.
OP may or may not be exposed to Irish VAT and / or VAT in the foreign country. My advice at this point is get proper advice OP, as failure to do so could be very costly down the line.