RemoveAndAddPoint

HTML

<script src="http://highcharts.com/js/testing.js"></script>
<div id="container" style="height: 400px"></div>

<button id="button1">Remove/Add point</button>

JavaScript

var chart = new Highcharts.Chart({
    chart: {
        renderTo: 'container'
    },

    xAxis: {
        categories: ['A', 'B', 'C', 'D']
    },

    series: [{
        data: [29.9, 71.5, 106.4, 129.2]}]
});

var i = 0;

$('#button1').click(function() {
    if (i % 2 == 0) {
        //remove point
        chart.series[0].data[2].remove();
        chart.xAxis[0].setCategories(['A', 'B', 'C']);
    } else {
        //add point
        chart.series[0].addPoint([2, Math.random() * 100]);
        chart.xAxis[0].setCategories(['A', 'B', 'C', 'D']);
    }
    i++;
});