I've been asked about details on the making of the movie. So here's some info I have gathered from several sources...
Statue of liberty: A scale replica of the upper 85 feet of Lady Liberty was constructed on location in Iztapalapa, Mexico (also called Ixtapalapa.) Accurate down to the rivets and patina, much of the scaffolding around the replica was the same scaffolding from the actual statue. No blue screen shots - it was all shot on the statue - or a replica, anyway.
First it was constructed from pipes and planks, then covered in a wire mesh screening, then sprayed with foam. The features were carved from the foam by ten Mexican sculptors. It was 1/4" smaller than the actual size - until a layer of 1/2" thick fiberglass was added. This was then hand-sanded by 25 men for three weeks. Then all the copper plates were attached and aged (painted.) Detailed photographs were used to get everything exactly right.
Total cost: $50,000. To make the same replica in the USA would have been over one million dollars.
Filming was even trickier. Remember - what you have is the top of the statue - so you can't film down. All you'd get would be a bisected Lady looking like she was sunk in dirt to her arm-pit. The solution: Shoot on the actual statue and the replica. All downward shots were don on the real statue before moving the production to Mexico. This was the only movie made on the Statue during it's face-lift.
Plus it was bizarre. Exit a highway near Mexico city, and in the distance you see... Lady Liberty.
The outdoor shots in the woods were also shot in Mexico, at an old Ski lodge in the Pass of Cortez. Shooting here lasted three weeks - at 13,000-14,000 altitude! There was also a cardio specialist on hand because of the extreme altitude. Some shots had to be moved to Mexico city because the altitude made in nearly impossible to work.
Interior shots were at Churubusco Studios in Mexico City. Part of the mountainside was reconstructed here. This is also where the interior shots of the Army base were shot. Take the scene where Remo and our soldier chick are following Grove - where winds up with Remo and our heroine trapped in the vacuum chamber. The chase tour like it winds through hundreds of feet of corridors, a dozen doors, twists and turns. In reality it was one corridor and three doors - all shot from different angles.
Even the East River shot of Sam Makin (pre-Remo) and the squad car taking a dive was shot on the lot. The car dropping into the water was shot in New York in November (COLD!). The underwater shots and Makin swimming from the car were done in Mexico in February. The use of the NYPD car required special permission from the Mexican Government - who would not let it out of the country after filming ended. It was left behind.
Now imagine getting permission to bring weapons into the country. The 60mm machine gun was verboten. So a replica was made. It's cheap to shoot in Mexico, but there's obviously lots of shortcomings! Perhaps they should have contacted Mexican rebels for the real thing...
Other shots: The ambulance driving through NY at night in a rain storm - real. The hospital room and doctor's dressing room - real (in an actual hospital.) Outside shot just before Remo meets Chiun: Exterior - NY. Interior - Mexico.