It's good that these things...
It's good that these things are hidden under the dash where no one can see them. More of you than not will have a heat-only box so yours might be simpler and lighter, but the procedures that apply will be the same for both. If you are pulling yours out of a car with an intact interior, it would be wise to pull the front seat and also protect your carpet from dripping antifreeze.
Have you ever noticed that there are some aspects of a musclecar restoration that are not very glamorous? I'm not talking about getting dirty rebuilding your engine or being coated in dust as you prep your car for paint. There are rewards for successfully completing those jobs. A well-detailed, strong engine or flawless, baby-butt smooth paint will bring you a lot of praise for your efforts.
But when was the last time you heard anyone remark about how great someone's heater box looks . . . probably never. But I'll bet you appreciate a properly working A/C on those mid-summer afternoons, or a little heat on a late-season, early morning cruise. If nothing else, you'd simply like your heater core to not drip antifreeze on your fresh carpet. No, it's not a glamorous job, but it's oh so much easier (read actually possible) if you take the time to pull the whole assembly out of the car.
This heater/air-conditioning box is out of a '68 Barracuda and is part of our progressive restoration series. The guys at Muscle Car Restorations (MCR) have done a lot of these over the years and have few tips that will save you some time and frustration.

The heater-core cover fiberglass...

The heater-core cover fiberglass gets brittle with age; if you just slip a screwdriver behind the clip to pry if off, you'll break the little ridge that holds the clip in place. Instead, use the screwdriver as shown to pry down and lift the edge of the clip up and then out.

The hoses are color coded...

The hoses are color coded so take careful notes on the routing. The same goes for the wiring. Taking detailed photos is a really good idea. You'll need to test the vacuum actuators and make sure they work in both directions-they should push and pull. New ones are not available so you'll have to track down good used ones if any of yours are bad.

Don't be surprised to find...

Don't be surprised to find broken fiberglass, especially around the support rod that attaches to the cowl vent opening. MCR uses a two-part bumper repair product: Fusor 142 Extreme Bumper Repair Adhesive from the Lord Corporation. It's strong, flexible, and sandable. Don't be afraid to be generous with the adhesive; this repair doesn't need to be pretty because it's hidden inside the box.

With the cover off, you can...

With the cover off, you can remove the screws that retain the heater core.