How to fix a not working Twitter push notification?
Part 1: iPhone, iPad, iPod (Do the procedure on all of your devices.)
1. Delete Twitter app.
2. Go to Settings > Twitter > Select your username > Delete account.
3. Reboot your device.
Part 2: Desktop
1. Go to www.twitter.com and sign in your account.
2. Go to www.twitter.com/settings/applications
3. You must see a list of applications that can access your twitter account.
4. Click 'Revoke Access' on ALL of the apps.
5. Refresh the page and you must not see any single app on the page.
Part 3: iPhone, iPad, iPod (Do the procedure on all of your devices.)
1. Go to Settings > Twitter > Add account
2. Sign in your account.
3. Install the Twitter app.
PS: Sorry for my english, I'm from the Philippines.