You can swap the dash without removing the heater box. Here is a write-up it did on changing the dash: (NOTE: After taking apart my 2nd gen parts car I found that you can unhook the dash harness under the hood and push it bach through the firewall. This would make the job a bit easier to to as you could leave the harness alone. )
It took me around 6 hours to change my dash.
You should have your new dash clean and ready to go. I would let it sit in the sun for a few hours to lessen the chance of cracking it while installing it, now for the fun:
1) This is optional but suggested: Remove the front seats, it only take a few minuets but it will give you tons more room to work. After the seats are out, disconnect the battery.
2) Remove the center console. Older style: 2 screw up front, 3 under the access panel in the rear. Newer style: 1 in cupholders, several inside glove box. Pop off console top and there are 2 more at the dash.
3) Optional: remove shifter.
4) Remove all the stuff from the glove box and remove the glove box (3 or 4 screw under the dash) and then remove the metal panel under the steering column (4 screws)
5) Remove the brace under the steering column (2 bolts). Disconnect the wiring harness to the column, remove the 4 nuts that hole the column guard, remove the 4 bolts that hold the column up and then disconnect the pinch bolt from the shaft that goes to the steering rack. Remove the column from the car. You will also need to remove the large bolt to the left of the column opening.
6) Remove the "A" pillar upper and lower trim. Pull carpet up on passenger side and disconnect the dash harness at the kick panel. Do the same at the driver’s side.
7) Remove the dash bezel and remove all switches, radio, EATC, sensors, and instrument cluster. Don't forget the ones on top of the dash and the modules and relays under the dash. Unbolt the fuse panel from the hinged arm. Take pictures and/or label the connections as you unhook them.
8) Remove the screws that hold the dash to the car: 3 at the windshield, 1 behind the radio, and 1 at each door jam. On 2nd gen cars there is one to the left of the floor heater duct.
9) Lift up and back on the dash, it should come off the firewall enough that you can unhook any other connection that you could not reach before.
10) Unbolt the main harness from behind the instrument panel frame. Unhook the wiring from the stereo from the main harness, the stereo harness can stay on the dash.
11) Carefully detach the main harness from the dash, taking note of how it is run through the dash.
12) Remove the dash. You should now have a big mess of wiring like this:
13) Clean EVERYTHING while you can.
14) Reinstall in reverse order. Be extremely careful with the new dash and they will crack easily when not in the car.
Have fun!