Correct Answer:

function addDays(date, days) {
  var result = new Date(date);
  result.setDate(result.getDate() + days);
  return result;
}

Incorrect Answer:

This answer sometimes provides the correct result but very often returns the wrong year and month. The only time this answer works is when the date that you are adding days to happens to have the current year and month.

// Don't do it this way!
function addDaysWRONG(date, days) {
  var result = new Date(); // not instatiated with date!!! DANGER
  result.setDate(date.getDate() + days);
  return result;
}

Proof / Example

Check this JsFiddle

// Correct
function addDays(date, days) {
    var result = new Date(date);
    result.setDate(result.getDate() + days);
    return result;
}

// Bad Year/Month
function addDaysWRONG(date, days) {
    var result = new Date();
    result.setDate(date.getDate() + days);
    return result;
}

// Bad during DST
function addDaysDstFail(date, days) {
    var dayms = (days * 24 * 60 * 60 * 1000);
    return new Date(date.getTime() + dayms);    
}

// TEST
function formatDate(date) {
    return (date.getMonth() + 1) + '/' + date.getDate() + '/' + date.getFullYear();
}

$('tbody tr td:first-child').each(function () {
    var (this);
    var ('<td/>').insertAfter($in).addClass("answer");
    var $outFail = $('<td/>').insertAfter(outDstFail = $('<td/>').insertAfter($outFail);
    var date = new Date($in.text());
    var correctDate = formatDate(addDays(date, 1));
    var failDate = formatDate(addDaysWRONG(date, 1));
    var failDstDate = formatDate(addDaysDstFail(date, 1));

    $out.text(correctDate);
    $outFail.text(failDate);
    $outDstFail.text(failDstDate);
    $outFail.addClass(correctDate == failDate ? "right" : "wrong");
    $outDstFail.addClass(correctDate == failDstDate ? "right" : "wrong");
});

body {
    font-size: 14px;
}

table {
    border-collapse:collapse;
}
table, td, th {
    border:1px solid black;
}
td {
    padding: 2px;
}

.wrong {
    color: red;
}
.right {
    color: green;
}
.answer {
    font-weight: bold;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<table>
    <tbody>
        <tr>
            <th colspan="4">DST Dates</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>03/10/2013</td></tr>
        <tr><td>11/03/2013</td></tr>
        <tr><td>03/09/2014</td></tr>
        <tr><td>11/02/2014</td></tr>
        <tr><td>03/08/2015</td></tr>
        <tr><td>11/01/2015</td></tr>
        <tr>
            <th colspan="4">2013</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2013</td></tr>
        <tr><td>02/01/2013</td></tr>
        <tr><td>03/01/2013</td></tr>
        <tr><td>04/01/2013</td></tr>
        <tr><td>05/01/2013</td></tr>
        <tr><td>06/01/2013</td></tr>
        <tr><td>07/01/2013</td></tr>
        <tr><td>08/01/2013</td></tr>
        <tr><td>09/01/2013</td></tr>
        <tr><td>10/01/2013</td></tr>
        <tr><td>11/01/2013</td></tr>
        <tr><td>12/01/2013</td></tr>
        <tr>
            <th colspan="4">2014</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2014</td></tr>
        <tr><td>02/01/2014</td></tr>
        <tr><td>03/01/2014</td></tr>
        <tr><td>04/01/2014</td></tr>
        <tr><td>05/01/2014</td></tr>
        <tr><td>06/01/2014</td></tr>
        <tr><td>07/01/2014</td></tr>
        <tr><td>08/01/2014</td></tr>
        <tr><td>09/01/2014</td></tr>
        <tr><td>10/01/2014</td></tr>
        <tr><td>11/01/2014</td></tr>
        <tr><td>12/01/2014</td></tr>
        <tr>
            <th colspan="4">2015</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2015</td></tr>
        <tr><td>02/01/2015</td></tr>
        <tr><td>03/01/2015</td></tr>
        <tr><td>04/01/2015</td></tr>
        <tr><td>05/01/2015</td></tr>
        <tr><td>06/01/2015</td></tr>
        <tr><td>07/01/2015</td></tr>
        <tr><td>08/01/2015</td></tr>
        <tr><td>09/01/2015</td></tr>
        <tr><td>10/01/2015</td></tr>
        <tr><td>11/01/2015</td></tr>
        <tr><td>12/01/2015</td></tr>
    </tbody>
</table>

In my humble opinion the best way is to just add a full day in milliseconds, depending on how you factor your code it can mess up if you are on the last day of the month.

For example Feb 28 or march 31.

Here is an example of how I would do it:

var current = new Date(); //'Mar 11 2015' current.getTime() = 1426060964567
var followingDay = new Date(current.getTime() + 86400000); // + 1 day in ms
followingDay.toLocaleDateString();

Imho this insures accuracy

Here is another example. I do not like that. It can work for you but not as clean as example above.

var today = new Date('12/31/2015');
var tomorrow = new Date(today);
tomorrow.setDate(today.getDate()+1);
tomorrow.toLocaleDateString();

Imho this === 'POOP'

So some of you have had gripes about my millisecond approach because of day light savings time. So I'm going to bash this out. First, Some countries and states do not have Day light savings time. Second Adding exactly 24 hours is a full day. If the date number does not change once a year but then gets fixed 6 months later I don't see a problem there. But for the purpose of being definite and having to deal with allot the evil Date() I have thought this through and now thoroughly hate Date. So this is my new Approach.

var dd = new Date(); // or any date and time you care about 
var dateArray =  dd.toISOString().split('T')[0].split('-').concat( dd.toISOString().split('T')[1].split(':') );
// ["2016", "07", "04", "00", "17", "58.849Z"] at Z 

Now for the fun part!

var date = { 
    day: dateArray[2],
    month: dateArray[1],
    year: dateArray[0],
    hour: dateArray[3],
    minutes: dateArray[4],
    seconds:dateArray[5].split('.')[0],
    milliseconds: dateArray[5].split('.')[1].replace('Z','')
}

Now we have our Official Valid international Date Object clearly written out at Zulu meridian. Now to change the date

dd.setDate(dd.getDate()+1); // this gives you one full calendar date forward
tomorrow.setDate(dd.getTime() + 86400000);// this gives your 24 hours into the future. do what you want with it.