Should I Replace Heat Exhanger in HVAC Unit When I Replace Compressor?

This is a good opportunity to upgrade. After 24 years, your coil is hopelessly clogged with dust. Easier to change than effectively clean. Consider a remote exchanger, incorporating a way to pre-heat or pre-cool fresh incoming air. This will improve your overall efficiency. At the same time, talk to your installer about a setup that is easy to access for cleaning. So many coils are almost impossible to clean.

Some installers just want to get the system up and working. Efficiency and ease of servicing are not their priority. (Get it in, get the check, get out)

You probably wont have much choice since the new refrigerants are often not compatible with the older ones and many HVAC people won't hook up a new compressor/refrigerant system to the older remains of a non-compatible system. You might want to check that out since it could make the decision for you.

These components are usually paired as a system for performance, and an old A-coil may not be conducive to optimum performance....at 24 years the coil has exceeded it's design life anyway(ie: you've got your money's worth out of it), so yes I would replace the evaporator coil at this time.....and thermostat....and possibly have duct work cleaned and inspected/repaired....

Did the compressor actually go bad and burn out or did it just forma leak at one of the brazed joints?

If it's just a bad joint I would simply re braze it and recharge the system.

At 24 years old it most likely a common R-22 refrigerant in the system which you yourself can still legally buy on line despite what your local heating and cooling weasels may tell you. I get my 30 cylinders off of eBay no question asked.

As far as cleaning old A coil that rather easy too if you have it out of the ducting. Just give it a good soaking with degreaser and wash it down with a garden hose from both sides the blow it dry with compressed air.

At worse yo may need to go over it with a long bristle wire brush. Just make sure you sweep back and forth in the same grain direction as the fins or you will be needing to buy a fin comb to fix them.

Granted buying a 30# cylinder of R-22 for $350 may sound expensive but going by what your local heating and cooling weasels are going to charge you do do what you yourself can legally do its a cheaper route.

Also a home central air conditioning system is considered a sealed split unit and fall under the same installation regulations that a window AC unit falls under which means legally you can take your old system out repair it and or replace it or any part of it yourself without having broken any laws if you so chose to do so.

That's my $1.50 on the situation.