How many Irish servants were shipped to America?

An exact figure for this question is not really possible to express because the records that would be necessary are either incomplete, unavailable, or poorly kept. We can work in estimates though: we know that the first English settlement at Roanoke was established in 1587 and that, by 1590, it had been abandoned. English migration to North America was slow at first, largely because it required hard work with unpredictable economic gains, a lack of consistent transportation, and the need to pay high transportation fees up front. The poor could not migrate to North America easily at this point, and the wealthy had little reason to without the likelihood of additional riches and at least some infrastructure.
This situation changed somewhat in 1606 when Virginia was granted its royal charter. The first settlers in Jamestown were mostly willing settlers. All were male, and some were boys (which makes the willingness difficult to ascertain), and all were from England (this does not preclude the inclusion of Irish individuals, though they would have been living in England at the time). During the first several years of the Jamestown settlement, life was difficult. Starvation was an almost regular occurrence, and hostilities between the colonists and the Native Americans caused many deaths. The colonists themselves did a poor job of providing sufficient food for their settlement, as most of the effort was placed on the recovery of natural resources. However, the colony began to reorient itself in 1616 with a focus on the growing of tobacco.
During the 10 years between the granting of the charter and the reorientation of the economic activity, women had migrated to the Virginia colony to be wives to the settlers. In addition, more planters had gained access to land because of the Virginia Company's inability to repay investors the previous year. This increased the amount of land that was being worked and the number of people. It also created a more permanent settlement, as families were now beginning to put down roots. The first African slaves arrived in 1619.
The record is inadequate to determine whether indentured servants were also present at this time. However, a notable influx of indentured servants to Virginia was recorded in 1630. This makes it likely that the total number of indentured servants was low prior to this time. It also seems likely that most of the individuals working land in the Virginia colony were English at this time.
The first notable influx of Irish migrants was recorded in 1715 and included about 20,000 migrants. The total Irish migration to North America between 1700 and 1775 has been estimated at a high of about 200,000. Assuming lower levels of migration between 1620 (when more colonies were formed) and 1700, it would be safe to estimate that approximately 30,000 additional Irish may have arrived, bringing a total number to 230,000. Some estimates (such as Bielenberg in The Irish Diaspora) have estimated that the total net Irish migration to North America and the West Indies between 1630 and 1775 was 165,000, with approximately 40,000 going to the West Indies. Using these two estimates, it would be reasonable to work off of a total gross migration figure to North America of 200,000 (this includes those who migrate to America and return to Ireland) between 1630 and 1775. This likely overestimates the number, but it is plausible.
Unfortunately, this 200,000 number includes voluntary migrants, which would seem to be outside the category of those shipped as servants. A large proportion of the Irish migrants came as indentured servants, which should not be surprising, as Irish migrants tended to be poor, largely because of English measures meant to disadvantage non-Protestants. Estimates of the indentured servitude place it as high as 50% among immigrants in general and as high as 65% among Irish immigrants. This would place the total number of Irish indentured servants between 100,000 and 130,000.
However, indentured servants were, largely, voluntary migrants that agreed to servitude in exchange for passage to North America. It would be difficult to describe these individuals as "shipped" over as their voyage was voluntary. However, included in this figure are indentured servants who were exiled to North America because of a conviction. These individuals were forcibly shipped to North America to serve as servants (in order to pay off passage and any other debts) and would fit within the question.
Estimates of the number of convicts shipped to North America by England come to about 52,000. It is unclear whether this includes prisoners of war (of which Cromwell sent 10,000–12,000 around 1650, though some may have been sent to Barbados) and other individuals captured and sold. Assuming it does not, we may add about 10,000 to account for individuals fraudulently sold as servants and those sent strictly to North America as prisoners of war (the Irish were generally sent to the West Indies or Leeward Islands because of a concern that Irish Catholics would ally with French Catholics against the English in North America). This brings us to 62,000 total convict servants. These would also include English convicts, so the number will have to be reduced. The Irish likely made up no more than 50% of the penal transports, though a range of 45%-55% is likely to be safe. This places the range between 27,900 and 34,100.