If the event has a cut-off date for registration, for example, event is on Saturday but cut-off date to register is Wednesday, maybe a banner or highlighted section showing this?
That's a good idea! Different from the "this event is in the past" alert.
But when we consider the fact that there are often multiple registration deadlines and other kinds of alerts such as "call for proposals," I think the solution lies in the event updates feature. The event update should have a "highlight this" option where highlighted event updates appear on the listing or site.
I just added to that feature a link to this one. Once we finish the updates feature, we can then do this one, where the event update can be highlighted. Other current event updates or past ones will go under the regular "Event Updates" section of an event.