Fixes oEmbedding tweets after Twitter's changes in Jan 2014. To use, download by clicking the "Download Gist" button below, upload to your WordPress site and activate. You will then need to re-save any posts edited or published since the 17 January.

composer.json

{
    "name": "cftp/twitteroembed-hotfix",
    "description": "Fixes oEmbedding tweets after Twitter's changes in Jan 2014",
    "license": "GPL-2.0+",
    "type": "wordpress-plugin",
    "authors": [
        {
            "name": "Tom J Nowell",
            "email": "[email protected]",
            "homepage": "http://tomjn.com"
        },
        {
            "name"    : "Code For The People",
            "homepage": "http://codeforthepeople.com/"
        }
    ],
    "require": {
        "composer/installers": "~1.0"
    }
}

twitteroembed-hotfix.php

<?php
/*
Plugin Name: Twitter OEmbed Hotfix
Description: Fixes oEmbedding tweets after Twitter's changes in Jan 2014
Author: Tom J Nowell
Version: 1.0
License: GPLv2 or later
License URI: http://www.gnu.org/licenses/gpl-2.0.html
*/

add_filter( 'oembed_providers', 'twitter_hotfix' );

function twitter_hotfix( $providers ) {
	$providers['#https?://(www\.)?twitter\.com/.+?/status(es)?/.*#i'] = array( 'https://api.twitter.com/1/statuses/oembed.{format}', true  );
	return $providers;
}