Emily,Here's one approach using the method Jeff suggested above (disable identify when measure tools are active). In the doIdentify function I check to see if any of the measurement tools have been selected. If they are then the identify is not performed. 
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<html>
  <head>
    <meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
    <meta http-equiv="X-UA-Compatible" content="IE=7,IE=9" />
    <!--The viewport meta tag is used to improve the presentation and behavior of the samples 
      on iOS devices-->
    <meta name="viewport" content="initial-scale=1, maximum-scale=1,user-scalable=no"/>
    <title>Identify Sample</title>
    
    <link rel="stylesheet" type="text/css" href="http://serverapi.arcgisonline.com/jsapi/arcgis/2.6/js/dojo/dijit/themes/claro/claro.css">
    <script type="text/javascript">djConfig = { parseOnLoad:true }</script>
    <script type="text/javascript" src="http://serverapi.arcgisonline.com/jsapi/arcgis/?v=2.6"></script>
    <style>
       html, body { height: 98%; width: 98%; margin: 0; padding: 5px; }
    </style>
    <script type="text/javascript">
      dojo.require("esri.map");
      dojo.require("esri.tasks.identify");
      dojo.require("dijit.layout.ContentPane");
      dojo.require("dijit.layout.TabContainer");
      dojo.require("dijit.form.Button");
      dojo.require("esri.dijit.Measurement");
 
      var map, identifyTask, identifyParams, symbol;
      var layer2results, layer3results, layer4results;
      var measurement; 
      
      
      function init() {
        var initExtent = new esri.geometry.Extent({"xmin":-9270392.071487641,"ymin":5247043.693206084,"xmax":-9269914.340060795,"ymax":5247401.99177622,"spatialReference":{"wkid":102100}});
        map = new esri.Map("mapDiv",{extent:initExtent});
        dojo.connect(map, "onLoad", initFunctionality);
        var basemap = new esri.layers.ArcGISTiledMapServiceLayer("http://server.arcgisonline.com/ArcGIS/rest/services/World_Street_Map/MapServer");
        map.addLayer(basemap);
        
        var landBaseLayer = new esri.layers.ArcGISDynamicMapServiceLayer("http://sampleserver3.arcgisonline.com/ArcGIS/rest/services/BloomfieldHillsMichigan/Parcels/MapServer",{opacity:.20});
        dojo.connect(map.infoWindow, "onShow", function() {
          dijit.byId("tabs").resize();
        });
        map.addLayer(landBaseLayer);
      }
      function initFunctionality(map) {
      
      
         measurement = new esri.dijit.Measurement({
            map: map
          }, dojo.byId('mdiv'));
          
 
          measurement.startup();
                
        dojo.connect(map, "onClick", doIdentify);
        identifyTask = new esri.tasks.IdentifyTask("http://sampleserver3.arcgisonline.com/ArcGIS/rest/services/BloomfieldHillsMichigan/Parcels/MapServer");
        identifyParams = new esri.tasks.IdentifyParameters();
        identifyParams.tolerance = 3;
        identifyParams.returnGeometry = true;
        identifyParams.layerIds = [0,2];
        identifyParams.layerOption = esri.tasks.IdentifyParameters.LAYER_OPTION_ALL;
        identifyParams.width  = map.width;
        identifyParams.height = map.height;
        
        map.infoWindow.resize(415, 200);
        map.infoWindow.setContent(dijit.byId("tabs").domNode);
        map.infoWindow.setTitle("Identify Results");
        symbol = new esri.symbol.SimpleFillSymbol(esri.symbol.SimpleFillSymbol.STYLE_SOLID, new esri.symbol.SimpleLineSymbol(esri.symbol.SimpleLineSymbol.STYLE_SOLID, new dojo.Color([255,0,0]), 2), new dojo.Color([255,255,0,0.25]));
      }
      function doIdentify(evt) {
    
      var measureMode =   dojo.query(".esriButton .dijitButtonNode").some(function(node, index, arr){
          if(node.childNodes[0].checked){
            //at least one of the measure tools is active so disable identify
            return true;
            
          }
       });
       if(! measureMode){
        map.graphics.clear();
        identifyParams.geometry = evt.mapPoint;
        identifyParams.mapExtent = map.extent;
        identifyTask.execute(identifyParams, function(idResults) { addToMap(idResults, evt); });
      }  
       }
        
     
      function addToMap(idResults, evt) {
        bldgResults = {displayFieldName:null,features:[]};
        parcelResults = {displayFieldName:null,features:[]};
        for (var i=0, il=idResults.length; i<il; i++) {
          var idResult = idResults;
          if (idResult.layerId === 0) {
            if (!bldgResults.displayFieldName) {bldgResults.displayFieldName = idResult.displayFieldName};
            bldgResults.features.push(idResult.feature);
          }
          else if (idResult.layerId === 2) {
            if (!parcelResults.displayFieldName) {parcelResults.displayFieldName = idResult.displayFieldName};
            parcelResults.features.push(idResult.feature);
          }
        }
        dijit.byId("bldgTab").setContent(layerTabContent(bldgResults,"bldgResults"));
        dijit.byId("parcelTab").setContent(layerTabContent(parcelResults,"parcelResults"));
        map.infoWindow.show(evt.screenPoint, map.getInfoWindowAnchor(evt.screenPoint));
      }
      function layerTabContent(layerResults, layerName) {
        var content = "";
        switch (layerName) {
          case "bldgResults":
            content = "<i>Total features returned: " + layerResults.features.length + "</i>";
            content += "<table border='1'><tr><th>ID</th><th>Address</th></tr>";
            for (var i=0, il=layerResults.features.length; i<il; i++) {
              content+="<tr><td>"+layerResults.features.attributes['PARCELID']+" <a href='#' onclick='showFeature(" + layerName + ".features[" + i + "]); return false;'>(show)</a></td>";
              content+="<td>"+layerResults.features.attributes['Full Site Address']+"</td>";
            }
            content+="</tr></table>";
            break;
          case "parcelResults":
            content = "<i>Total features returned: " + layerResults.features.length + "</i>";
            content += "<table border='1'><tr><th>ID</th><th>Year Built</th><th>School District</th><th>Description</th></tr>";
           for (var i=0, il=layerResults.features.length; i<il; i++) {
              content+="<tr><td>"+layerResults.features.attributes['Parcel Identification Number']+" <a href='#' onclick='showFeature(" + layerName + ".features[" + i + "]); return false;'>(show)</a></td>";
              content+="<td>"+layerResults.features.attributes['Residential Year Built']+"</td>";
              content+="<td>"+layerResults.features.attributes['School District Description']+"</td>";
              content+="<td>"+layerResults.features.attributes['Property Description']+"</td>";
            }
            content+="</tr></table>";
            break;
        }
        return content;
      }
      function showFeature(feature) {
        map.graphics.clear();
        feature.setSymbol(symbol);
        map.graphics.add(feature);
      }
      dojo.addOnLoad(init);
    </script>
  </head>
  <body class="claro">
    Click the map to identify building and tax information.
    <div id="mapDiv" style="width:800px; height:600px; border:1px solid #000;"></div>
    <!-- info window tabs -->
    <div id="tabs" dojoType="dijit.layout.TabContainer" style="width:385px;height:150px;">
      <div id="bldgTab" dojoType="dijit.layout.ContentPane" title="Buildings"></div>
      <div id="parcelTab" dojoType="dijit.layout.ContentPane" title="Tax Parcels"></div>
    </div>
    <div id='mdiv'>My test div area</div>
  </body>
</html>