So what is the process you envisage?
I go to your website.
I see a print I like.
I purchase it on your website.
You send the order to "Drop Shipping Ltd" in the UK
They print it.
they post it to Ireland or the UK or America.
I doubt that they would be interested for a small level of business as the costs of setting up would outweigh the potential profits.
If you are shipping to Ireland (or the EU) where I presume your base is, there is no VAT if you are not registered for VAT. Postage would be cheaper. So I reckon you should ship Irish orders yourself.
If a large UK company ships to Ireland, the Irish customer will have to pay 13.5% VAT and higher postage. I presume that they can look after the VAT themselves so that the customer in Ireland won't have to go to the hassle of paying An Post or Fedex for it. They will pay the VAT at the point of purchase. But it just seems an additional expense to the customer and you will have to pay Drop Shipping Ltd for the service. And what if it doesn't arrive? They will contact you via your website.
There are photo agencies you could use for non EU customers e.g. Getty Images. Ah, but they probably just license the use of the photo which is probably not what you want.
Have you a link to a drop shipping company which does what you want it to do?
You should speak to fellow art photographers who can tell you how they cracked this problem.