It is offensive in a number of ways.
The Irish are too stupid to know what twins really are.
The Irish are too backwards to have access to contraception.
Yes, exactly, I fail to see why even the people who are rushing on to the thread to proclaim it's a phrase they have always used without thinking so it must be fine cannot see this once it's pointed out to them. It's simply not a factual way of registering two siblings born very close together, it's specifically associating that biological fact with a network of negative, stereotypical connotations to do with Irish people being feckless, ignorant and backward.
This is the Urban Dictionary explanation:
The roots of the idea behind the term are actually quite old, although no one knows when, exactly, people first began to talk about Irish twins. In both England and the United States, a massive influx of Irish immigration in the 1800s led to a negative connotation with Irish people and society. This often happens when a large immigrant group begins to settle in mass numbers in a new country. The Irish were accused of being backwards and uncultured, and it was assumed that they were uneducated, dirty, and a general pox on society. As a result, the use of the word “Irish” began to be pejorative.
A number of derogatory terms incorporating stereotypes about the Irish began to emerge, including “Irish confetti” for thrown bricks and “Irish kiss” for a slap. Irish twins fits into this vernacular, and is actually insulting on multiple levels.
Firstly, the term pokes fun at the stereotypical fertility of Irish Catholic families, which traditionally did not use birth control. In addition, it implies that the Irish lack the ability to plan ahead or control themselves, having children in quick succession rather than responsibly spacing them. Finally, it suggests that the Irish do not understand the medical definition of twins, which involves two children conceived and born together.