Although there are a few different kinds of cherries, most of them are not very acidic and will therefore not give you much heartburn at all after eating them. California cherries for example have an acidity rating of around 4.01 which is considered to be mildly acidic but not substantially so. Frozen cherries on the other hand are considered to be very acidic and can very well produce problems with heart burn should you consume enough of them.
Black canned cherries are considered to be mildly acidic and therefore not much a problem for those who have problems with heart burn/acid reflux. There are also Maraschino cherries which have an acidity rating which falls around 3.47 which is considered to be very acidic and should be avoided by those who have serious problems with acid reflux.
Regular red cherries are considered to be extremely acidic, so if you know that you have a problem with acidic reflux, you should probably forgo eating them altogether. The fact is that those who have acid reflux problems can still eat cherries, but they will need to be careful about how much they eat and also which kind they consume, because both things will matter quite a bit.