Este código funciona perfecto. El problema era que el botón estaba dentro de un panel de actualización de control y eso causaba algún problema. Pero ya lo arreglé y ahora funciona.

 Dim documentoPDF As New Document(PageSize.A4, 10, 10, 10, 10)
        Dim memStream As MemoryStream = New MemoryStream()
        Dim writer As PdfWriter = PdfWriter.GetInstance(documentoPDF, memStream)
        documentoPDF.Open()

        Dim Paragraph1 As Paragraph = New Paragraph("First paragraph")
        documentoPDF.Add(New Paragraph(Paragraph1))

        documentoPDF.Close()

        Dim bytesInStream As Byte() = memStream.ToArray()
        memStream.Close()

        'Dim memorystreaam As MemoryStream = New MemoryStream(bytesInStream)

        Response.Clear()
        Response.ContentType = "application/pdf"
        Response.AddHeader("content-disposition", "attachment;filename=FileName.pdf")
        Response.Buffer = True
        Response.Cache.SetCacheability(HttpCacheability.NoCache)
        'Response.OutputStream.Write(memStream.GetBuffer(), 0, memStream.GetBuffer().Length)

        Response.BinaryWrite(bytesInStream)
        Response.End()