I also started experiencing this problem a month or two ago (May/June 2012).
I'm on a Motorola Droid 4
Android 2.3.6
System 6.13.219.XT894.Verizon.en.US
Location: Springfield, MO area.
Signal Level: 4 of 5 bars.
Plan: Family Shared 900 with grandfathered Unlimited Data from the Alltel buyout.
Some days the 4G LTE works fine, some days it will occasionally not work, other days it simply won't work at all. The issue isn't limited to the Facebook App, but any app that accesses the internet. Interestingly when streaming music via Pandora it never seems to stop playing even when the connection is down, perhaps Pandora does a good job at buffering enough a head of time, otherwise the data outage is just limited to NEW connections and not existing data flows.
Same symptoms as posted here.
Will have BLUE 4G LTE signal, as soon as an app is opened that accesses data the signal indicator turns WHITE or GREY and the app says "Data Connection Lost" or "No Data Connection" or the webpage will timeout. Sometimes I can wait a while and it will start working, other times I just put it into Airplane Mode for a second then turn the radios back on and it temporarily resolves the issue for a few minutes.
I doubt it is my phone because I have a family member with the DROID RAZR and they have the exact same problem. The internet is full of this exact complaint yet when I've tried to contact Verizon support they don't seem to be aware of the issue and simply want me to keep factory restoring my phone...which hasn't fixed the problem yet and I don't have days and days to argue with them anymore. I haven't even been able to reach a verizon support rep that knows what the BLUE versus WHITE/GREY signal indicator means.
The only fix I have found is to disable 4G and only let the phone use 3G, it is rock solid and never looses connection.
To do this, go to Settings, --> Wireless & Networks, --> Mobile Networks, --> Network Mode, --> select CDMA only.