Are you referring to the furnace or the evaporation coil?
Aaren Assuming you are talking about a new condensing unit for an AC there are a few of good answers.
1) Yes if the old indoor coil is from a low efficiency unit and was designed for the same size unit as the new outdoor unit. That is the only way you are going to get the rated efficiency from the new unit
2) No if the old indoor coil is clean and is say from a 4 ton unit, but you are putting in a 2 or 2.5 ton outdoor unit. If you also change the metering device to a properly sized TXV then you can come up with a really sweet efficient AC.
3) Better yet, if the only problem is the compressor of the old 4 ton unit, put in a new 2 or 2.5 ton compressor and a new properly sized TXV.
Yes it works
Be sure to check/clean the indoor coil before reusing. Many coils have been in the system for years and are clogged up with dirt. Other than that it's just tubes and fins.
If you do the smaller condenser while keeping large coil, check airflow. You may need to adjust speed or replace blower motor to get correct CFM.
Also note that you won't be eligible for any rebates unless the coil is changed.
What about issues of refrigerant incompatibility?
Refrigerant is all gone once system is opened up. There is ongoing debate about oil compatibility.
Hopefully Bob was saying the refrigerant was recovered not: "all gone" into the atmosphere.
If you have a R22 unit any replacements will have to be R22 equipment. If you are replacing everything you will be putting in R-410A and the refrigerant oils are not compatable. The old oil will need to be cleaned from the lineset unless you replace it too. Don't mix the oils -- gunk will result.
I have taken a 5 ton coil Air handler and 3 tons of duct work and sealed the home and added R and went back with a 2.5 ton AC system keep the old but in good shape coil and air handler end up with a 15.3 EER with a 15.5 SEER unit. I used the old line set and flushed with r-11 flush and lots of presser - nitrogen. Then Vack to 300 Mic. Its works well for 3 yr but is not ARI