In early medieval Ireland, the correct dating of Easter was something that put the local church at odds with the rest of the western European Christian church. Other religions did and do have celebrations that fall on different days every year, like the Ramadan for the Muslims. Are there examples in history of religious controversies caused by different interpretations of the calendar?
So the first thing that you have to understand is that the Arab calendar, pre-Islam, was a 12 month lunisolar calendar. The first day of the new month began when you could see the first sliver of the waxing crescent, the night after a new moon. They had an extra, intercalatory month that they used to fill in the extra time and ensure that they fit to a full seasonal year.
In the year 10 of the Islamic calendar, or 631-632 AD, the intercalatory month was prohibited. There is some debate as to why, but the important thing is that the calendar switched from lunisolar to pure lunar. So, the months started "rotating" backwards through the seasons. Now what's important to note is that the date of Ramadan never changed, rather, the actual system of the calendar did. From the perspective of a lunar calendar, it's the same time every year.
Now, that's not to say there are no controversies about the calendar. However, they're of a slightly different nature. People argue incessantly as to where the moon should be spotted to determine the beginning of the month. There are two main camps: the Mecca camp who believe the month starts when you spot the waxing crescent in Mecca, and the hometown camp, who believe that it's whenever you physically see the waxing crescent looking at the sky. There are good points to both sides and degrees of each, though I find myself leaning towards the hometown approach. Note, these are not official names, I don't know if there even are official names.
That being said though, controversy is a strong word. The arguments, while spirited, are short lived, and almost entirely limited to the few days surrounding the first day of Ramadan. It's hard to exactly explain the significance of it, but rest assured that nobody is getting killed, excommunicated, or otherwise ostracized from the community for their opinion on the matter.