I tried to adapt your code to mine but it didn't work. The same problem still exists, as if the macro couldn't detect the 2 quotation marks.
Here's my updated code:
Sub MoveToBeginningSentence()
Application.ScreenUpdating = False
Selection.Cut
Selection.MoveLeft Unit:=wdSentence, Count:=1, Extend:=wdMove
Selection.MoveRight Unit:=wdCharacter, Count:=1, Extend:=wdExtend
If Selection.Characters.First Like "[ " & Chr(34) & Chr(147) & Chr(11) & vbCr & vbTab & ".!?]" Then
Selection.MoveLeft Unit:=wdSentence, Count:=1, Extend:=wdMove
Selection.MoveRight Unit:=wdCharacter, Count:=1, Extend:=wdMove
Selection.Paste
Selection.MoveRight Unit:=wdCharacter, Count:=1, Extend:=wdExtend
Selection.Range.Case = wdLowerCase
Selection.MoveLeft Unit:=wdSentence, Count:=1, Extend:=wdMove
Selection.MoveRight Unit:=wdCharacter, Count:=1, Extend:=wdMove
Selection.MoveRight Unit:=wdCharacter, Count:=1, Extend:=wdExtend
Selection.Range.Case = wdUpperCase
Else
Selection.MoveLeft Unit:=wdSentence, Count:=1, Extend:=wdMove
Selection.Paste
Selection.MoveRight Unit:=wdCharacter, Count:=1, Extend:=wdExtend
Selection.Range.Case = wdLowerCase
Selection.MoveLeft Unit:=wdSentence, Count:=1, Extend:=wdMove
Selection.MoveRight Unit:=wdCharacter, Count:=1, Extend:=wdExtend
Selection.Range.Case = wdUpperCase
End If
End Sub