There is a definite spike in the number of applications for Irish passports from British people. But we won't the full extent of this for some time.
The Belfast Telegraph reports that 7,045 locals joined the June rush for Irish passports. This sounds like a lot, but it only an extra 614 applications from the same time last year (9.54%^). The total number of applications for Irish passports for June was 78,408, so the increase is less than 1%. Not huge.
Interestingly the increase in applications from Great Britain is much higher, both in raw numbers and percentages. Namely, 1048 and 21.99%.
These figures come from the official stats page on the Irish Governemnt's Department for Foreign Affairs and Trade website.
Interestingly there's an article in the Irish Mirror entitled Thousands of British people have applied for Irish passports since Brexit vote that quotes very similar figures based on a Freedom of Information request to the Irish Government. Further more it says they received 2,300 forms from the North from June 23 – the date of the referendum – to June 30, and 1,475 from Great Britain.
Bear in mind that's only one week, and most people will take longer than that to get all the paperwork, referees, photos, birth certificates etc sorted out. I know it did when I did it :-). (Brexit Day 16: Irish passport application sent.) I think the figures for July will be far more interesting.
Source - Irish Mirror: Thousands of British people have applied for Irish passports since Brexit vote