We have some XML which effectively represents the contents of a table. Each row from the table is an element; each column in a row is a
subelement within the row. The XML we generate for the table could
contain more columns than we actually want to process/display.

I have another element which identifies all of the columns to be
processed/displayed. The subelements of this element have the same
names as the row columns.

I'm looking for recommendations on ways in which we can only process the
row columns which correspond to the other columns.

XSL (fragments) w/ filtering::

<xsl:variable name="TableColumns" select="/Table/Columns/*" />

<xsl:template  select="/Table/Row">
   <xsl:for-each select="*">
      <xsl:variable name="MyName" select="name()" />
      <xsl:if test="$TableColumns[name()=$MyName]" >
         <xsl:apply-templates select="." mode="ColumnContent" />
      </xsl:if>

Instead of selecting all elements and filtering out later, select
only the elements you want. Also, using a key might speed it up:
 <xsl:key name="cols" match="/Table/Columns/*" use="name()"/>
 <xsl:template  select="/Table/Row">
    <xsl:for-each select="*[key('cols',name())]">

I also suggest to try other XML designs, like
<Table>
  <Columns>
    <Column>1</Column>
    ...
  </Columns>

  <Row>
    <Column id="1">...</Column>
    ...