First off the carrier heater units come with a wiring diagram.
You stated this
The instructions on the new heater says to install the grey, orange, and blue wires in 1-2-3 of the blower tap connection. Our furnace has C-1-2-3-4. Do I put the wires in C-1-2 or skip C and put them in 1-2-3?
Now think about what you said it said to do. It said to install the wires in 1,2,3, of the blower tap. Now think real hard it did not say anything about C or 4 did it.
Did you research this heater and is it the correct one for your unit to start with? Carrier is really good when it comes to its instructions so I am sure they did not make a mistake.