visa

You can stay in Ireland for three months after your student visa expires, provided you can satisfy airport immigration that you have sufficient funds for your stay and that you will indeed be going home at the end of the three months (a return ticket to the US is essential).

Other European countries and those that are part of the Schengen zone will have different rules. There isn't a single Europe-wide visa policy.

Best thing to do is to contact your Embassy/consulate.

Most likely you have a schengen-zone visa, which includes Poland but not Ireland (or the UK). You should have no problems entering Ireland (presuming you are US-citizen), but you could have problems leaving and re-entering the schengen area if you outstayed your visa.
On possible solution would be to go to Ireland before your visa expires (which gives you up to 90 days in Ireland) and then return to the schengen area after (starting off your 90 day period in the schengen zone). However; don't take my word for it (or that of any casual poster here); contact your embassy!

By the way, i have never heard of the requirement to have a return ticket to the US to enter Ireland.

Do a Google search on Americans refused entry at Dublin Airport for not having a return ticket... you're sure to get a number of hits.

Do a Google search on people being abducted by aliens and you get many more: If you have a legitimate reason not to have a return ticket (eg studying in Poland) i doubt you will need one.
Then again; another question to ask the right people (eg. Irish embassy in Poland).

I take it you didn't do that search, then.

Well if OP wants to try to enter Ireland with no (unexpired) visa to return to another country and no return ticket back to the US, that is up to OP. Hopefully s/he won't be foolish enough to assume s/he won't need one just because someone on an internet forum, whose nationality means he has no need for one himself, hasn't ever heard of other people needing them.

Again: The OP needs to contact his/her Embassy/Consulate (and the Irish one in Poland) to sort out this complicated visa issue (which i have been saying in my initial post), and not listen to any advice given on this forum (by me or anyone else).

If you have pre-paid for your Ireland studies AND you have a ticket back to the States to show any immigration people who ask, you should be fine.
I'd suggest flying to the UK, as then you will definitely have records of 'leaving Schengen (and Poland, whoch excluded from Schengen for you as long as you are on a Polish student visa)', then making your own way to Ireland, and when Ireland is done....you should be free to travel where you like for as long as you like, assuming you are on a US passport and have a ticket back to the US at the end of summer....