First, don't write queries in a for loop.
Second, you can save all the items needed to be updated in a list, then update them.
Thirdly, if I understand correctly, you need to update Opportunity field based on the OpportunityLineItem field, right?
Here is the code:

trigger Projectrollup on Opportunity (after insert, after update) 
{
  Map<Id,Opportunity> oppIds = new Map<Id,Opportunity>();
  List<Opportunity> updateList = new List<Opportunity>();

  for(Opportunity o : Trigger.new)
  {
    oppIds.put(o.id,o);
  }

  OpportunityLineItem[] OLI = [Select UnitPrice, Quantity, PricebookEntry.Product2Id, TotalPrice,PricebookEntry.Product2.Name, Description, 
                                Converted_to_Asset__c,Asset__c,PricebookEntry.Product2.Create_Asset__c, PricebookEntry.Product2.Sales_Order_Group__c
                               From OpportunityLineItem 
                               where OpportunityId IN : oppIds.keySet() And PricebookEntry.Product2.Product_Reporting_Category__c = 'Services'];

  for(OpportunityLineItem o : OLI)
  {
    if(o.PricebookEntry.Product2.Sales_Order_Group__c = 'Hotline Services' || o.PricebookEntry.Product2.Sales_Order_Group__c = 'Policy Management Services' || o.PricebookEntry.Product2.Sales_Order_Group__c = 'Incident Management Services')
    {
      Opportunity opp = new Opportunity(id=o.OpportunityId);
      opp.Implementation_Team__c = true;
      updateList.add(opp);
    }
    if(o.PricebookEntry.Product2.Sales_Order_Group__c = 'ReadyTraining Services')
    {
      Opportunity opp = new Opportunity(id=o.OpportunityId);
      opp.Training_Team__c = true;
      updateList.add(opp);
    }
  }

  if(updateList.size()>0)
  {
    update updateList;
  }
}


***:In the first for loop, MAYBE you need to add some line like if o.HasOpportunityLineItem== true, oppIds.put(o.id,o); depends on the real situation.   Let me know if you still have any questions ~^_^