Here is an example of a procedure that removes top 15 lines from any text file (regardless of the contents of those lines). It should be able to handle arbitrarily large files as it works on one line at a time.
Sub removeTopLines()
    Dim srcFile As String, nSrc As Integer  ' source file (read from)
    Dim dstFile As String, nDst As Integer  ' destination file (write to)
    Dim textLine As String
    Const LINES_TO_SKIP = 15
    Dim lineCounter As Long
    srcFile = "c:\opt\src.txt"
    dstFile = "c:\opt\dst.txt"
    nSrc = FreeFile
    Open srcFile For Input As #nSrc
    nDst = FreeFile
    Open dstFile For Output As #nDst
    lineCounter = 1
    Do Until EOF(nSrc)
        Line Input #nSrc, textLine
        If lineCounter > LINES_TO_SKIP Then
            Print #nDst, textLine
        End If
        lineCounter = lineCounter + 1
    Loop
    Close #nDst
    Close #nSrc
End Sub
You can see an example of how to traverse a directory tree here, or alternatively you can just get a list of all those file path names and call this procedure from another loop giving it one file at a time.
Update: Here is another version that instead of a line count looks for a string that contains "time" and copies only the lines after that.
Sub removeTopLinesAfter()
    Dim srcFile As String, nSrc As Integer  ' source file (read from)
    Dim dstFile As String, nDst As Integer  ' destination file (write to)
    Dim textLine As String, strAfter As String
    strAfter = "time"
    Dim copyLines As Boolean
    srcFile = "c:\opt\src.txt"
    dstFile = "c:\opt\dst.txt"
    nSrc = FreeFile
    Open srcFile For Input As #nSrc
    nDst = FreeFile
    Open dstFile For Output As #nDst
    copyLines = False
    Do Until EOF(nSrc)
        Line Input #nSrc, textLine
        If Not copyLines Then
            copyLines = InStr(textLine, strAfter) > 0
        Else
            Print #nDst, textLine
        End If
    Loop
    Close #nDst
    Close #nSrc
End Sub