The Roman Catholic Church is fast running out of priests. In Ireland in 2007, 160 Catholic priests died, while only nine new recruits were ordained. To say the least, those figures don't point towards sustainability.
There are two things of importance to note in response to this:
(1) The Catholic Church does indeed have a serious shortage of priests at present and for the foreseeable future.
(2) The shortage is not due to "running out of priests".
There were some declines in the two decades of the eighties and the nineties; but at present the number of priests is actually growing, and has grown for several consecutive years, albeit slowly. The Irish numbers are irrelevant to the matter, at least by themselves, because anyone who knows anything about the demographics of the priesthood knows that not all priests even in Ireland are Irish. The Catholic Church is a global organization; if numbers decline in one part of the world, this can be made up for by priests from another part of the world. And this is exactly what one finds: Europe and North America have (in many, but not in all, places) had sharp declines, Asia and Africa have (again, in many places) had considerable surges, and the Third World surges are currently balancing, and slightly overtopping, First World declines to create a net growth in the number of priests worldwide. This does mean that the age of the ubiquitous Irish priest, which has given so much of an Irish flavor to Catholicism all over the world, is past, even in Ireland; the new Irish, so to speak, are priests from places like Uganda. And seventy-five years from now when people think of a likely name for a Catholic priest, it won't be 'Patrick O'Malley' or anything like it; it will be something more like 'Ambrose Sentamu', and he'll have a clipped Ugandan accent rather than an Irish brogue.
The real problem with the shortage is that, due to serious unsustainability in the First World (looked at on its own), the African and Asian growth is just barely keeping things at growth level, while the Catholic population is, in comparison, exploding. The Catholic population is keeping steady with global population growth (and has been for several decades now) -- that is, despite significant global population growth, Catholics have continued to be around about 17% of the world population. But the population of priests has recently undergone a period of decline in a number of areas of the world, and its overall current growth is much, much smaller than the growth of the Catholic population at large. Africa and Asia cannot produce priests fast enough to fulfill an increasing global demand. The result is that priests are increasingly scarce, and parishes without a priest are increasingly common. Catholic church services that are not Masses are already beginning to be widespread, and are likely to become more and more common as priests are spread more thinly. And deacons, who can, except for some of the sacraments, do everything a priest can, are becoming increasingly important, and the number of those is steadily increasing.
There are changes in store, but they are not really more startling than any other changes; there is no danger at present of the Catholic Church "running out of priests". It's slowly gaining priests, in fact. It just isn't gaining them fast enough to keep pace with the massive expansion of the Catholic population. (Somehow, I suspect, Dawkins would be less pleased with this than he is with the "running out of priests" idea.) In any case, it's a good example of why you should be cautious about focusing on a region or country when dealing with something that extends beyond it: data about Ireland may be merely anecdote about the world.
CARA is an interesting resource for those interested in numbers dealing with Catholic demographics.