In the worksheet "Monthly Service", change C1 to a number, ie. Jan would be 1, Feb would be 2, etc. Then apply this code

Option Explicit

Sub NextService()
    Dim ws As Worksheet, sh As Worksheet
    Set sh = Sheets("Monthly Service")
    Dim i As Long, lr As Long, lrM As Long
    lrM = sh.Range("B" & Rows.Count).End(xlUp).Row
    sh.Range("B4:E" & lrM + 1).ClearContents
    Application.ScreenUpdating = False
    Application.Calculation = xlCalculationManual
    For Each ws In Worksheets
        If ws.Name <> "Monthly Service" Or ws.Name <> "CONDITIONS" Then
            lr = ws.Range("H" & Rows.Count).End(xlUp).Row
            lrM = sh.Range("B" & Rows.Count).End(xlUp).Row + 1
            For i = 3 To lr
                If Month(ws.Range("I" & i)) = sh.Range("C1") Then
                    ws.Range("H" & i).Copy
                    sh.Range("B" & lrM).PasteSpecial xlPasteValues
                    ws.Range("B1").Copy sh.Range("C" & lrM)
                    ws.Range("C" & i).Copy sh.Range("D" & lrM)
                    ws.Range("A" & i).Copy sh.Range("E" & lrM)
                End If
            Next i
        End If
    Next ws
    lrM = sh.Range("B" & Rows.Count).End(xlUp).Row
    Range("B4:B" & lrM).NumberFormat = "m/d/yyyy"
    Application.CutCopyMode = False
    Application.Calculation = xlCalculationAutomatic
    Application.ScreenUpdating = True
    MsgBox "Action Complete"


End Sub