Be careful with the "Range(...)" without first qualifying a Worksheet because it will use the currently Active worksheet to make the copy from. It's best to fully qualify both sheets. Please give this a shot (please change "Sheet1" with the copy worksheet):
EDIT: edited for pasting values only based on comments below.
Private Sub CommandButton1_Click()
Application.ScreenUpdating = False
Dim copySheet As Worksheet
Dim pasteSheet As Worksheet
Set copySheet = Worksheets("Sheet1")
Set pasteSheet = Worksheets("Sheet2")
copySheet.Range("A3:E3").Copy
pasteSheet.Cells(Rows.Count, 1).End(xlUp).Offset(1, 0).PasteSpecial xlPasteValues
Application.CutCopyMode = False
Application.ScreenUpdating = True
End Sub
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