Below is my code after making some changes...
I tested it and most of the combinations are working... Just try it...
Code:
Private Sub Command1_Click()
 Dim StoreDaysTest As String
    Dim TheDay, FDay, LDay As String, TestDay As String
    Dim n As Integer
    
    StoreDaysTest = "Sun - Thu"
    TestDay = Mid$(StoreDaysTest, 1, 3)  ' get first 3 chars to represent day.
    FDay = Mid$(StoreDaysTest, 1, 3)
    LDay = Right(StoreDaysTest, InStr(1, StrReverse(StoreDaysTest), "-") - 2)
    Me.Caption = FDay & "=" & LDay
    n = 0
    Do
        TheDay = Format(Date + n, "DDD") ' increment day, return first 3 char.
        If Checker(TheDay) = Checker(FDay) Or Checker(TheDay) = Checker(LDay) Then 'Exit Do
            MsgBox Format(Date + n, "long date")
            Exit Do
        ElseIf Checker(TheDay) > Checker(FDay) And Checker(TheDay) < Checker(LDay) Then 'Exit Do
            MsgBox Format(Date + n, "long date")
            Exit Do
        'Else
        Else
            n = n + 1
        End If
    Loop
    
    'Debug.Print Format(Date + n, "long date")

End Sub

Private Function Checker(ByVal Temp) As Integer
If Temp = "Sun" Then
    Checker = 1
ElseIf Temp = "Mon" Then
    Checker = 2
ElseIf Temp = "Tue" Then
    Checker = 3
ElseIf Temp = "Wed" Then
    Checker = 4
ElseIf Temp = "Thu" Then
    Checker = 5
ElseIf Temp = "Fri" Then
    Checker = 6
ElseIf Temp = "Sat" Then
    Checker = 7
End If
End Function
-Best of luck
Akhilesh