[SalesForce] “Non-selective query against large object type” only occurs in trigger context despite filtering on lookups

The error message is quite detailed, the query needs to be more selective so that less data is returned. There's a lot of into on that online, and an answer on how to solve this can't really be given without detailed knowledge of your object, its fields and specific query use case.

Yet, in general you should not use != in soql because it can't utilize any table indexes and will require a full table scan. To start optimizing your query, consider changing

GE_HQ_CONTACT_STATUS__c != 'Inactive' 

into

GE_HQ_CONTACT_STATUS__c IN ('Active','Other Status','Yet Another Status')

I think your Order object contains more than 200,000 records. That's why it is complaining.

Secondly you are using SOQL inside a for loop, that's also need to be avoided.

More Efficient SOQL Queries

For best performance, SOQL queries must be selective, particularly for queries inside triggers. To avoid long execution times, the system can terminate nonselective SOQL queries. Developers receive an error message when a non-selective query in a trigger executes against an object that contains more than 200,000 records. To avoid this error, ensure that the query is selective.

Selective SOQL Query Criteria

• A query is selective when one of the query filters is on an indexed field and the query filter reduces the resulting number of rows below a system-defined threshold. The performance of the SOQL query improves when two or more filters used in the WHERE clause meet the mentioned conditions.

• The selectivity threshold is 10% of the first million records and less than 5% of the records after the first million records, up to a maximum of 333,333 records. In some circumstances, for example with a query filter that is an indexed standard field, the threshold can be higher. Also, the selectivity threshold is subject to change.

Refer Working with Very Large SOQL Queries

Approach will be like this:

Also it be recommended to create a Trigger handler to put your processing logic.

trigger StartFlow on Order__c (after update ) 
{
    Set<Id> opptyIds = new Set<Id>();
    public Flow.Interview.Create_Renewal_OppLineItems_from_Invoice_record DummyFlow {get; set;}
    if(Trigger.isAfter && Trigger.isUpdate)
    {
        for(Order__c o : Trigger.New) 
        {
            //put the comparison criteria for records to be filtered.
            opptyIds.add(o.Opportunity_Name__c);        
        }
        List<Order__c> Ord =[SELECT ID,Account_Name__r.Customer_Profile_Pricebook__c,End_Date__c,Year_Enddate_1_day__c,
                                 Journal_Reader_Code__c,Number_of_Users__c,Renewal_Number__c, Opportunity_Name__c, 
                                 Product__r.Id,Delegate_Admin__c,CurrencyIsoCode 
                                 FROM Order__c 
                                 WHERE Opportunity_Name__c IN:opptyIds];
        if(Ord.size()>0)
        {
            string[] value0 = new string[]{Ord[0].Account_Name__r.Customer_Profile_Pricebook__c};
            date[] value1 = new date[]{Ord[0].End_Date__c};
            double[] value2 = new double[]{Ord[0].Year_Enddate_1_day__c};
            string[] value3 = new string[]{Ord[0].Journal_Reader_Code__c};
            double[] value4 = new double[]{Ord[0].Number_of_Users__c};
            double[] value5 = new double[]{Ord[0].Renewal_Number__c};
            Id[] value6 = new Id[]{Ord[0].Opportunity_Name__r.Id};
            Id[] value7 = new Id[]{Ord[0].Product__r.Id};
            string[] value8 = new string[]{Ord[0].Delegate_Admin__c};
            string[] value9 = new string[]{Ord[0].CurrencyIsoCode};

            Map<String, Object> myMap = new Map<String, Object>();
            myMap.put('VarCustomerProfile', value0);//Customer_Profile_Pricebook__c    
            myMap.put('VarInvoiceEnddate', value1);  //End_Date__c
            myMap.put('VarInvoiceEnddateYear', value2);  //Year_Enddate_1_day__c
            myMap.put('VarInvoiceJRC', value3); //Journal_Reader_Code__c
            myMap.put('VarInvoiceNumberOfUsers', value4); //Number_of_Users__c
            myMap.put('VarInvoiceRenewalNumber', value5);//Renewal_Number__c
            myMap.put('VarOppId', value6);//OpportinityID
            myMap.put('VarProductId', value7);//ProductId
            myMap.put('VarDelegateAdmin', value8);//Delegate_Admin__c
            myMap.put('VarInvoiceCurIsoCode ', value9);//CurrencyIsoCode

            DummyFlow = new Flow.Interview.Create_Renewal_OppLineItems_from_Invoice_record(myMap);
        }
    }
}