Hi Puja,

I feel there is better way to solve the error except post my code, please check below. 

I worried that I shouldn't use add the Contact Owner ID directly. 

Thank you so much.

trigger updateClassOwner on Contact (after update) {

	String mailTemplateId = [SELECT id FROM EmailTemplate WHERE Name = 'updateClassOwner'].id;
	//To get the Id from Trigger
	Set<Id> classIds = new Set<Id>();
	//The Id for sending Email
	List<ID> targetObjectIds = new List<ID>();
	List<ID> whatIds = new List<ID>(); 
	
	for(Contact fellow: trigger.new)
	{
		classIds.add(fellow.Id);
	}
	
	List<Class__c> fellowClass = new List<Class__c>([SELECT id, OwnerID FROM Class__c WHERE fellow__c in: ClassIds]);
	List<Class__c> updateClass = new List<Class__c>();
	
	for(Contact fellow: trigger.new)
	{
		//compare if contact's owner is changed.
		Contact oldOwner = Trigger.oldMap.get(fellow.Id);
		If(oldOwner.OwnerId != fellow.OwnerId)
		{
			targetObjectIds.add(fellow.OwnerId);
			whatIds.add(fellow.Id);
			//query all the class that this fellow is teaching
			for(integer i = 0; i < fellowClass.size(); i++)
			{
				//update Class' OwnerId
				fellowClass[i].OwnerId=fellow.OwnerId;
				updateClass.add(fellowClass[i]);
			}
		}
	}
		
	update updateClass;
	If(targetObjectIds != null)
	{
	    Messaging.MassEmailMessage mail = new Messaging.MassEmailMessage();
	    mail.SetTargetObjectIds(targetObjectIds);
	    mail.setTemplateId([SELECT id FROM EmailTemplate WHERE Name = 'updateClassOwner'].id);
	    mail.setWhatIds(whatIds);
	    mail.saveAsActivity = false;
	    Messaging.sendEmail(new Messaging.MassEmailMessage[] { mail }); 
	}
}

Thank you so much for help.