Is it possible to turn off the iPhone Calendar invitations badge/list?

Solution 1:

Actually, this functionality has been moved into the Notification settings for Calendar items (Settings > Notifications > Calendar).

In that view you'll have the option to change notifications for: Upcoming Events; Invitations; Invitee Responses; and Shared Calendar Changes. Each one give the ability to select: whether its shown in notification center; notification sound; badge app icon; show on lock screen; and the alert style!

Solution 2:

There are two things that you can do here.

1. Open Settings > Mail, Contacts, Calendars > Scroll down to the Calendars section (near the bottom), and change "New Invitation Alerts" to Off.
2. In the Settings app, tap Notifications > select Calendars, and then turn off Calendars notifications in Notification Center (if desired), and set the Alert style to "None", and, most importantly, turn "Badge App Icon" to Off.

Unfortunately, to the best of my knowledge, there is no way to specifically disable the Red badge that is inside the app. However, it is possible turning off "New Invitation Alerts" will have that effect.