test.php

<?php

# create DOM instance
$dom = new DOMDocument();
$dom->load('test.xml');

# create XPATH instance of DOM
$xpath = new DOMXPath($dom);

# define xpath query to date value
$datePath = './bar/date';

# export all childs from root to array
$items = iterator_to_array($dom->documentElement->childNodes);

# filter out Text Nodes
$items = array_filter($items, function(DOMNode $current) {
    return ! $current instanceof DOMText;
});

# sort items by date value
usort($items, function(DOMNode $a, DOMNode $b) use ($datePath, $xpath) {
    $dateElements['a'] = $xpath->query($datePath, $a);
    $dateElements['b'] = $xpath->query($datePath, $b);

    $date['a'] = DateTime::createFromFormat('d.m.Y', trim($dateElements['a'][0]->nodeValue));
    $date['b'] = DateTime::createFromFormat('d.m.Y', trim($dateElements['b'][0]->nodeValue));
    
    if ( $date['a'] == $date['b'] ) {
        return 0;
    }
    
    return $date['a'] < $date['b'] ? -1 : 1;
});

# apply running index to the sorted array
$ordered = array_values($items);

# define the search object
$searchFor = DateTime::createFromFormat('d.m.Y', '24.12.2014');

# search
$result = null;
foreach ( $ordered as $key => $element ) {
    $current = DateTime::createFromFormat('d.m.Y', trim($element->nodeValue));
    if ( $current < $searchFor || $current > $searchFor ) {
        continue;
    }
    
    $result = compact('key', 'element');
}

if ( empty($result) ) {
    echo 'nothing found.';
}
else if ( $result['key'] > 0 ) {
    echo 'previous date of '.$searchFor->format('d.m.Y').' is '.trim($ordered[ $result['key'] - 1 ]->nodeValue);
}
else {
    echo 'search value was found, but has no previous date';
}

text.xml

<?xml version="1.0" encoding="utf-8" ?>
<data>
    <foo>
        <bar>
            <date>01.04.2014</date>
        </bar>
    </foo>
    <foo>
        <bar>
            <date>03.09.2014</date>
        </bar>
    </foo>
    <foo>
        <bar>
            <date>04.05.2014</date>
        </bar>
    </foo>
    <foo>
        <bar>
            <date>13.02.2014</date>
        </bar>
    </foo>
    <foo>
        <bar>
            <date>24.12.2014</date>
        </bar>
    </foo>
    <foo>
        <bar>
            <date>19.03.2014</date>
        </bar>
    </foo>
    <foo>
        <bar>
            <date>23.11.2014</date>
        </bar>
    </foo>
    <foo>
        <bar>
            <date>01.08.2014</date>
        </bar>
    </foo>
</data>