Do you see any error messages in Thunderbird?

Not that I am aware of. SOGo Connector usually display these overlays saying how many contact have been synced and I have not seen any message there until I rebooted the computer. At that point SOGo Connector re-downloaded the duplicates and notified me of that.

Is there a place where Thunderbird writes logs to?

See in Thunderbird under "Extras" --> "Error console Ctrl+Shift+J"
Do you see any errors in Thunderbirds error console when removing the duplicates?

I went through the process of removing a few hundred duplicates and there was no error in the console:

• "Tools" --> "Error Console (Ctrl+Shift J)"
• Cleared the console
• "Tools" --> "Address Book (Ctrl+Shift B)"
• "Tools" (in the Address Book window, not in the main window. Thunderbird 31.4.0 on Xubuntu 14.04 is using confusing terminology!) --> "Find and Manage Duplicates..."
• selected the same remote address book on the left and right drop downs, ticked the box to first collect duplicates, then clicked start to get the process going
• deleted a few hundred duplicates: NOTHING IN THE CONSOLE.

Then I right-clicked the address book's name on the left panel of the Address Book window and selected "Synchronize" from the context menu. Now I got a FLURRY OF ERRORS, all of which looked the same (I redacted identifiable information from the report):

Upload failure uploading card: https://REDACTED_SERVER_URL/REDACTED_PATH/C6797C85-CB70-0001-2E2E-C43011901AF1.vcf.
HTTP Status Code:412

• card properties:
  0 prop: [xpconnect wrapped nsIProperty]; name: DisplayName; value: REDACTED
  1 prop: [xpconnect wrapped nsIProperty]; name: groupDavKey; value: C6797C85-CB70-0001-2E2E-C43011901AF1.vcf
  2 prop: [xpconnect wrapped nsIProperty]; name: RecordKey; value: 25604
  3 prop: [xpconnect wrapped nsIProperty]; name: PreferMailFormat; value: 0
  4 prop: [xpconnect wrapped nsIProperty]; name: UID; value: 383be8c5-e86e-4eed-a4f7-6ece9f795b65
  5 prop: [xpconnect wrapped nsIProperty]; name: DbRowID; value: 304
  6 prop: [xpconnect wrapped nsIProperty]; name: groupDavVersion; value: -1
  7 prop: [xpconnect wrapped nsIProperty]; name: groupDavVersionPrev; value: "f0d89da070d86dac3960e8f5794f4244"
  8 prop: [xpconnect wrapped nsIProperty]; name: PopularityIndex; value: 1
  9 prop: [xpconnect wrapped nsIProperty]; name: PhotoType; value: generic
  10 prop: [xpconnect wrapped nsIProperty]; name: LowercasePrimaryEmail; value: REDACTED
  11 prop: [xpconnect wrapped nsIProperty]; name: FirstName; value: REDACTED
  12 prop: [xpconnect wrapped nsIProperty]; name: LastName; value: REDACTED
  13 prop: [xpconnect wrapped nsIProperty]; name: CardUID; value: C6797C85-CB70-0001-2E2E-C43011901AF1.vcf
  14 prop: [xpconnect wrapped nsIProperty]; name: PrimaryEmail; value: REDACTED
  15 prop: [xpconnect wrapped nsIProperty]; name: AllowRemoteContent; value: 0
  16 prop: [xpconnect wrapped nsIProperty]; name: SecondEmail; value: REDACTED
  17 prop: [xpconnect wrapped nsIProperty]; name: LowercaseSecondEmail; value: REDACTED
  18 prop: [xpconnect wrapped nsIProperty]; name: LastModifiedDate; value: 1423074443
  19 prop: [xpconnect wrapped nsIProperty]; name: PreferDisplayName; value: 0
• done

HTTP error code 412 Precondition Failed.
This should help the developers to identify the cause of your problem.

Could you check the Lightning version too?
You can find it under "Tools" --> "Add Ons" --> "Extensions"