Do I need to pay VAT on products I buy in another EU country?
YES. Duty-free shopping, for example in airports, is only for non-EU residents whose are travelling to a destination outside the EU. EU residents have to pay VAT, even when they shop in another EU country.
You may sometimes be exempt from paying VAT when you buy a car in another EU country.
I am an American citizen planning to travel to the EU later this year. If I buy goods during my stay in the EU, will I be able to claim back the VAT on those goods before I fly back to the USA?
YES. As long as you present the goods to customs on departure from the EU, together with the VAT refund documents, within 3 months of buying them.
Are there specific conditions for claiming a VAT refund in Ireland before I leave the EU?
YES. You have to prove you are resident outside the EU and you have to export the goods from the EU within 3 months of buying them. In some countries there is also a minimum value of purchases to qualify for VAT refunds, but this is not the case in Ireland.
I plan to travel from Paris to New York later this year. I know I can purchase duty-free goods in the tax-free shop at the airport, but is there a limit to what I can buy?
NO. When you're leaving the EU, there are no limits on the quantity or value of the duty/tax-free goods you can buy.
However, there may be a limit on what you can bring in to the country of destination (in this case the USA). You need to check this with the (US) authorities.