It all depends on the type of property and the type of request.

For external requests such as dining & theater, definitely call the concierge and speak with him. To the extent that the request is for something scarce, the sooner the better. To the extent that it is your first time dealing with the concierge, make sure he knows that you're going to take care of him quite nicely on arrival.

Internal requests really depend on the property. Is this truly a luxury property or a place that markets itself as 12-star but is just a chain hotel with a nice lobby? A true luxury property will be in touch with you. As others note, arrival time can really matter. Calling the day before to request early arrival may mean that the suite they had in mind won't be available because the guest checking out is holding it for a late checkout.

Airport transfers depend on the city. Hotels may well charge 6-7x the going rate for car service. That's fine if you are in a city where you don't speak the language and where people simply disappear. OTOH, at JFK where security rules don't allow cars to wait either curbside or unattended, you won't see a difference in wait time between Uber Black, DIAL7 or one of the expensive services.