Lightning: Get sobject tab icon

I know it is very late, but below solution worked for me for Standard and custom object.

Apex Class

    @AuraEnabled
    public static String getIconName(String sObjectName){
        String u;
        List<Schema.DescribeTabSetResult> tabSetDesc = Schema.describeTabs();
        List<Schema.DescribeTabResult> tabDesc = new List<Schema.DescribeTabResult>();
        List<Schema.DescribeIconResult> iconDesc = new List<Schema.DescribeIconResult>();

        for(Schema.DescribeTabSetResult tsr : tabSetDesc) { tabDesc.addAll(tsr.getTabs()); }

        for(Schema.DescribeTabResult tr : tabDesc) {
            if( sObjectName == tr.getSobjectName() ) {
                if( tr.isCustom() == true ) {
                    iconDesc.addAll(tr.getIcons());
                } else {
                    u = 'standard:' + sObjectName.toLowerCase();
                }
            }
        }
        for (Schema.DescribeIconResult ir : iconDesc) {
            if (ir.getContentType() == 'image/svg+xml'){
                u = 'custom:' + ir.getUrl().substringBetween('custom/','.svg').substringBefore('_');
                break;
            }
        }
        return u;
    }

Lightning Component

<aura:component controller="yourApexController" implements="flexiPage:availableForAllPageTypes,force:hasSObjectName">
    <aura:handler name="init" value="{!this}" action="{!c.doInit}"/>
    <aura:attribute name="sObjectName" type="String" />
    <aura:attribute name="iconName" type="String" />
    <lightning:icon iconName="{!v.iconName}" size="large" alternativeText="Indicates approval"/>
</aura:component>

Lightning JS controller

({
    doInit : function(component, event, helper) {
        var action = component.get("c.getIconName");
        action.setParams({ "sObjectName" : component.get("v.sObjectName") });
        action.setCallback(this, function(response) {
           component.set("v.iconName", response.getReturnValue() );
        });
        $A.enqueueAction(action);
    }
})

Let me know if it works for you.

For any of those who may be looking for an even more generic solution, or possibly a use case where you are simply using the SLDS styles in a VF page, I have a method which returns the lightning style class for the wrapper of the tab icon, in addition to its Lightning png icon to fill it. This is supported in a managed package, and will also pull custom icons that are defined by customers/packages installed. If an object is passed in that does not have a tab definition, it simply returns the SLDS Custom icon as a default. My use case required dynamic loading of the icon depending on a configuration for the object, so a dynamic IMG binding was much more realistic to switch up on the fly than repainting an SVG.

Method that takes in sObject Name and returns a map of properties (iconStyle, iconURL)

/***
 * Explores the schema of the soject passed in and finds lightning icon style and image url for sObject
 * If nothing is found, defaults to the custom icon
 *
 * @param   sObjectTypeName       the sObject to find the icon for
 * @return  Map<String,String>    Map of properties about the icon to be consumed by the front end.
 *                                Right now, returns values for keys: iconStyle,iconURL
 */
global static Map<String,String> retrieveIconForObject(String sObjectName) {    
    String iconStyle;
    String iconURL;
    String iconTag;
    Map<String.String>                iconPropertyMap = new Map<String.String>();
    List<Schema.DescribeTabSetResult> tabSetDesc      = Schema.describeTabs();
    List<Schema.DescribeTabResult>    tabDesc         = new List<Schema.DescribeTabResult>();
    List<Schema.DescribeIconResult>   iconDesc        = new List<Schema.DescribeIconResult>();

    for(Schema.DescribeTabSetResult tsr : tabSetDesc){
       tabDesc.addAll(tsr.getTabs()); 
    }
    for(Schema.DescribeTabResult tr : tabDesc) {
        if( sObjectName == tr.getSobjectName() ) {
            system.debug(tr.getIcons());
            if(!tr.getIcons().isEmpty()){
                iconDesc.addAll(tr.getIcons());
            }else{
                if( tr.isCustom() == true ) {
                    iconStyle = 'slds-icon-standard-custom';
                    iconURL   = '/img/icon/t4v35/standard/custom_120.png';
                } else {
                    iconStyle = 'slds-icon-standard-' + sObjectName.toLowerCase();
                    iconURL   = '/img/icon/t4v35/standard/' + sObjectName.toLowerCase() + '_120.png';
                }
            }
        }
    }
    for (Schema.DescribeIconResult ir : iconDesc) {
        if(ir.getTheme() =='custom' && !ir.getContentType().startsWith('image/svg')){//custom icon thats not an svg
            iconTag   = 'custom';
            iconURL   = ir.getURL();
            iconStyle = 'slds-icon-standard-custom';
            break;
        }else if(ir.getWidth() == 120 && ir.getTheme() =='theme4' && ir.getContentType() == 'image/png'){//large lightning icon image
            if(ir.getURL().contains('/custom/')){ //Icon is from the Custom Sprite
                iconTag    = ir.getURL().substringBetween('custom/','.png').substringBefore('_');
                iconURL    = '/img/icon/t4v35/custom/' + iconTag + '_120.png';
                iconStyle  = 'slds-icon-custom-'+ iconTag;
                break;
            }else if(ir.getURL().contains('/standard/')){//Icon is from the Standard Sprite
                iconTag    = ir.getURL().substringBetween('standard/','.png').substringBefore('_');
                iconURL    = '/img/icon/t4v35/standard/' + iconTag + '_120.png';
                iconStyle  = 'slds-icon-standard-'+ iconTag;
                break;
            }
        }
    }
    //if no icons are found, just default to the Custom lightning icon
    if(iconStyle == null){
        iconStyle = 'slds-icon-standard-custom';
        iconURL   = '/img/icon/t4v35/standard/custom_120.png';
        iconTag   = 'No Lightning Icon Found';
    }
    //return a map with our properties for the front end
    iconPropertyMap.put('iconStyle',iconStyle);
    iconPropertyMap.put('iconURL',  iconURL);

    return iconPropertyMap;
}

Example usage for binding:

<span data-bind="css: selectedMapObjectIconClass" class="slds-icon_container slds-avatar">
  <img data-bind="attr:{src: selectedMapObjectIconUrl}"></img>
</span>