I have list of Joda-Time Interval objects.
List<Interval> intervals = new ArrayList<Interval>();
How can I sort the intervals on the beginning Date of each interval. The intervals are not overlapping
Just create a Comparator<Interval> which compares by start times:
public class IntervalStartComparator implements Comparator<Interval> {
    @Override
    public int compare(Interval x, Interval y) {
        return x.getStart().compareTo(y.getStart());
    }
}
Then sort using that:
Collections.sort(intervals, new IntervalStartComparator());
In your special case, collect the start instants using
interval.getStart()
in another list.  DateTime using the Comparable interface which makes the list sortable using Collections.sort(..).