How indexing on a date field that is constantly updated and or omitted affect performance?
Hi It will affect but not greatly.what happens is whenever you made change in collection index will be updated and during index update your collection will be locked .So to overcome this you can use index in background  by using:
db.collection.createIndex( { last_updated: 1}, {background: true} ) 

it will let you use your collection while updating so you will never found any kind of disruption because of your index. 

As you have mentioned that you do sorting on "last_updated".I strongly recommend you to use index .you will have high read performance at a minimal indexing cost.