Your trigger is declared as before insert as well as before update.  However, insert triggers have no concept of previous values as the record is new to the database.  Therefore I'd imagine its your use of trigger.old that is the problem.  

How to fix this depends on your use case - if a new record indicates that the email has "changed", you can simply exclude new records from the tests:

 trigger CompOldNew on Opportunity (before insert,before update) {
    Map<Id,String> oppIDs=new Map<Id,String>();
   
    if (trigger.isUpdate)
    {
       for(Opportunity opp : Trigger.new){
                    //Create an old and new map so that we can compare values
                                   
                     Opportunity oldOpp= Trigger.oldMap.get(opp.ID);
                     Opportunity newOpp = Trigger.newMap.get(opp.ID);
            System.debug('oldOppoldOpp&&&&&&'+oldOpp+'newOppnewOpp$$$$$'+newOpp);
                    //Retrieve the old and new Reseller Email Field           
                    string oldResellerEmail = oldOpp.Reseller_Email__c;
                    string newResellerEmail = newOpp.Reseller_Email__c;
   System.debug('oldResellerEmail==='+oldResellerEmail+'newResellerEmail==='+newResellerEmail);           
                    //If the fields are different, the email has changed
                    if(oldResellerEmail != newResellerEmail){
                        oppIDs.put(opp.Id,opp.Reseller_Email__c);
                          
                    }
           System.debug('MAP VALUE======'+oppIDs);
       }
     }
     else
     {
         // new records - add all
         oppIds.addAll(trigger.new);
     }
                   

List<Opportunity> ls=new List<Opportunity>([select name,Reseller_Email__c from Opportunity where id in :Trigger.oldMap.keySet()]);
System.debug('XXXXXXXXXXXXXXXXXXXXXX'+ls);


}