The problem with your criteria is that you are specifying AND, and the conditions are mutually exclusive. A date can't be both earlier than 3/31/2013 and greater than 1/1/2014. (I'm not sure what happens if you mix AND and OR in the filter criteria.)
Try this change. AutoFilter does not have built-in filtering by quarter, although it does by month. You can specify each month to be included.
If SearchPeriod = "Q1" _
And SearchYear = "All" _
And SearchCategory = "All" _
And SearchProductLine = "All" Then
Sheets("Pipeline").ListObjects("PipelineTable").Range.AutoFilter 'Turns off any Auto Filters
ThisWorkbook.Sheets("Summary").Range("A6", Range("X" & Rows.Count).End(xlDown)).EntireRow.ClearContents 'Deletes contents from previous search
With Sheets("Pipeline")
.AutoFilterMode = False 'Auto Filters for all Q1 2013 and 2014 (01/01/2013-03/31/2013) records
Sheets("Pipeline").ListObjects("PipelineTable").Range.AutoFilter Field:=24, _
Operator:= xlFilterValues, Criteria2:=Array(1, "1/28/2013", 1, "2/28/2013", 1, "3/17/2013", 1, "1/28/2014", 1, "2/28/2014", 1, "3/17/2014")
ActiveSheet.Range("$A$1:$A$91").AutoFilter Field:=1, Operator:= _
xlFilterValues, Criteria2:=Array(1, "1/28/2013", 1, "2/28/2013", 1, "3/17/2013", 1, _
"10/29/2013", 1, "11/29/2013", 1, "12/26/2013")
ThisWorkbook.Sheets("Pipeline").UsedRange.Offset(1, 0).SpecialCells(xlCellTypeVisible).Copy
Worksheets("Summary").Range("A6").PasteSpecial xlPasteValues 'This copies-and-pastes the search results from the "Pipeline" to "Summary" tabs
End With
End If
ThisWorkbook.Sheets("Summary").Range("A6").Select
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