I've learned a lot being a Fixer and AP for production crews going to Easter Island. Here are some tips to make the planning process easier.
CARNET (I'm putting this one first because most productions do not give enough time or coordinate their flights to accomodate for Carnet processing at customs. Make sure you or your crew has enough connection time to get this done! Approximate 2-3 hrs for entry and 2-3 hrs for exit.)
If traveling from the US with production equipment you will need to travel with Carnet vouchers from the United States Council for International Business (USCIB). The process is long and at time tedious, but it is best to travel with one so that customs officials from either country don't try and slap you with an import/export tax. Chile is a Carnet country, and though they may not ask to see one once you enter or leave the country, you should notify officials at the customs counter (where the scanning devices are) that you are traveling with production equipment and will need your Carnet stamped. See the Baggage section for the process of getting this done in Santiago.
There are around one to two flights a day (and sometimes three) that come into Easter Island's international airport, Mataveri (IPC). These flights come in from Santiago's main airport, Arturo Merino Benitez (SCL), from Faaa Airport in Papeete (PPT), and occasionally direct from Lima, Peru (LIM). The only airline that flies into the island is Lan (Lan Chile or Lan Peru).
Flying In: Make sure your bags get ticketed all the way to IPC (Mataveri - Easter Island). If possible, try to fly with one of Lan's One World partners, in the US it's American Airlines. If you are traveling from the US or Europe into one of these airports and then passing through, you will most likely only be assed the baggage fee of the first airline you are flying in on. If your flight is interupted in Santiago, for example, you will need to pay Lan's additional overage fees. As of this writing (Nov 2011) Lan does not have a media rate, sorry.
Entering Chile through SCL with your Gear: You will need to pick up your bags in SCL and check them through customs. Notify the customs personel at the baggage scanning machines that you are a TV or Film crew and are traveling with an ATA Carnet. They are somewhat used to the process of filling it out and getting it stamped, but make sure they do it right. If you ask nicely enough, they will write in any additional comments that you think may resolve an issue when traveling back to your home country. At this point, because your bags are already ticketed for your next flight to IPC, politely ask the customs officials if they could send the bags back through the internal carrousel system so that they go directly to your flight.
Sometimes this works, but if it doesn't, you will need to take your bags up to the 3rd floor and get them rechecked in. At this point, Lan may try to charge you for any overages that you have. Make sure you bring documentation about any fees that you were already assessed at your home port.
Exiting Chile through SCL with your Gear: Have your fixer speak with the airport baggage personel if you have a lot of gear. You will most likely be instructed to arrive early with all of it including the passports and itineraries of all your crew. If you are flying on one of Lan's partners, the Easte Island desk will be able to check your bags through to your final port. Make sure you notify the person checking you in that you will need to pick up your bags in SCL to get your Exit stamp on your Carnet. At this point, you will be assessed Lan's overage fees.. though thankfully their international fees are lower than their domestic ones.
Once in SCL, repeat the process of picking up your bags taking it through customs so that you get your exit stamp, and having the customs official put your bags back on the internal system. At this point you should give yourself a high-five! You made it!
Picture above is the Mataveri International terminal. The island is remote. Get used to it.