If an Irish company sends workers to work in Poland can they operate them through the irish payroll system or does the company have to register as an employer in Poland and operate Polish PAYE on those employees?
Article 15 of the Ireland/Poland double tax treaty covers the taxation of employees.
If the employees are resident in Ireland, do not spend more than 183 days in Poland, the salary is paid by an employer not resident in Poland and the remuneration is not borne by a permanent establishment or a fixed base which the employer has in Poland then the wages would be only liable to Irish tax.
While srase is correct at the end of the process, it is worthwhile taking out some local advice on this. Polish law may require all sorts of procedural hoops to be jumped through even though the DTA 'cuts to the chase' in terms of whats applicable.
If an Irish company sends workers to work in Poland can they operate them through the irish payroll system or does the company have to register as an employer in Poland and operate Polish PAYE on those employees?
You need to get proper advice. I doubt Poland even have a PAYE system, I'd expect they have the typical European Federal/Local split with taxes being calculated at both levels, payable in arrears. In addition then there will be some kind of withholding system for expats, but since Irish workers are EU, it may be that it does not apply to them.
If you are going to have people their continuously, then you most likely will be subject to local commercial and labour market rules. Perhaps needing to set up pension funds etc...
I'd be inclined to agree about local advise as in the past I have experienced the Poles not accepting branches of foreign companies (as you may have in Poland). You may actually need to incorporate a Polish subsidiary. If you are putting Polish employees on the payroll of the Irish company you should obtain PAYE exclusion orders from the Revenue so that you don't have to operate PAYE. Generally speaking tax treaties don't overrule withholding tax requirements (which is what PAYE is).