Use the onPaste Callback

Use the onPaste option to define a callback that will strip the tags and manually insert the text.

$editor.summernote({
    onPaste: function (e) {
        var bufferText = ((e.originalEvent || e).clipboardData || window.clipboardData).getData('Text');
        e.preventDefault();
        document.execCommand('insertText', false, bufferText);
    }
});

Credit: https://github.com/summernote/summernote/issues/303

Solve Firefox Problems:

You may still have problems with Firefox. If so, wrap document.execCommand in a timer function to delay its execution a tiny bit:

setTimeout(function(){
    document.execCommand( 'insertText', false, bufferText );
}, 10);

Update for v0.7.0+

Position of callbacks in options is changed after v0.7.0
After v0.7.0, every callbacks should be wrapped by callbacks object. (source)

This means that the original code should be written as

$editor.summernote({
    callbacks: {
        onPaste: function (e) {
            var bufferText = ((e.originalEvent || e).clipboardData || window.clipboardData).getData('Text');
            e.preventDefault();
            document.execCommand('insertText', false, bufferText);
        }
    }
});

Credit to Mathieu Castets for pointing this out (so if this bit helped, upvote his answer!)

TL;DR: Here’s a functional demo