There are different limits for different API functions from what I have found out. Changing guild notes etc has a much lower limit than sending map pings for example.
As for the map pings, I did some extensive tests on the PTS and found that the limit is 2 calls per second over a prolonged time. But that is a gross oversimplification. Actually it is a bit more complicated. In the latest version of LibMapPings I added a leaky bucket implementation which drops any pings that would go over the limit and get you kicked.
But that implementation also doesn't account for the burst that Chip mentioned and there is also a different limit for in and out of combat:
As long as you stay below the curve, you won't get kicked and in the second image you can see that 2 pings per second is infinitely sustainable.
In general it would be great if ZOS could implement some functionality to prevent getting kicked from the game on the client side. For example a function to get the cooldown for the different functions and also warning events that a call has been dropped if an addon sent too many calls, instead of just dropping the player to the login screen.