In my 70, I had to remove the whole dash pad and glovebox. Once that stuff was out of the way, the rest of the job was easy.
Some advice.. Just cut the heater hoses at the firewall and buy new ones. Makes it easier to remove, they are cheap to replace, and the last thing you want it to do the job over again becasue a hose developed a leak.
Ya nitrous's idea is a good one, Also be careful when putting the hoses on the fittings to the heater core, the fittings and top of the heater core are not very strong, maybe a little silicon lube on the fitting will help it slide over easy, the reason i say this is i just replaced the heater core on my 85 f150 and i actually bent the fitting and the top tank trying to slide the hose over top.....
Im sure someone with a 71 will chime up here and give some more info...
1. Disconnect Heater and AC hoses. (See manual on proper AC disconnection procedures.)
2. Cut drain hose from under car at bottom of firewall. You may want to save the plug that’s inside the end of the drain hose, which is there to prevent critters from entering your heater box. Note: If you bypass this step, you will likely break the fiberglass drain nipple off your heater box trying to pull it out.
3. Remove glove box door and glove box insert.
4. If you have a camera, take a picture of the vacuum hose and temperature control cable area now before disassembly. Label hoses to further aid in reassembly.
5. Disconnect: Thermostat switch wires; vacuum hoses to heater box (2) vacuum canisters; vacuum hoses to (cable controlled) water valve switch; and cable to water valve switch.
6. On the engine side of the firewall, remove 2 heater box hex/washer nuts. Easily identified by large washer and long protruding studs.
7. Remove 2 blower motor housing retaining screws (under dash center) and permit blower housing to hang (no need to remove the firewall mounting on the blower motor).
8. You may need to remove the vacuum canisters to give the heater box assembly some wiggle room during removal and installation.
9. Remove the heater box to cowl retaining screw (look up at right side of heater box).
10. Remove heater box assembly by sliding to the right (to disengage the blower fan), then down.
11. On the now removed heater box, put a large screwdriver blade in the return of each metal spring clip such that you are opening up the springs to remove them. Do NOT pry the clips off by inserting the screwdriver under the clips – This will permanently damage the heater box.
Reverse to install.
Though some of the above is not included in the shop manual, if you need more info. (such as diagrams), I strongly recommend buying the full shop manual set.
While you have the heater box out, do a thorough cowl rust inspection as this is the first place these years rot out. If you're removing the heater box because you are getting water on your passenger side floor, it's more likely that you have cowl rust issues than a bad heater core. If you don't have any water issues, or coolant loss problems, and you're just not getting heat, you probably have a mouse nest blocking up the intake. You may want to wait with the purchase of a new heater core until you've fully diagnosed the problem. To prevent future "mouse nest" problems you may want to find a way to rig a screen over the cowl intake.