If the added structure is well done and does not have any safety issues, then more than likely they will not make an issue of it.
Since you purchased this home you can ask the inspector to see what it would take to have it permitted. Since it was done back in the 1960's then I am sure many things will have to be brought up to code in order for it to be retro permitted.
Who would look for it?
The odds are atronomical!
In addition, the Company that you hire to do the A/C and Heater will be getting the PERMIT, no you.
Just don't stand there with a guilty look when the Inspector comes.
And don't tell anyone else.
If your added room was a patio that was enclosed it may be very obvious that it was done without permits. However in general if it looks old, and there haven't been any complaints, they don't investigate past permit status. I suspect this is intentional. They want to have work done with permits. If they go searching for problems, homeowners would be more reluctant to get permits fearing turning a small project into a major headache. Also building codes change. Few old homes meet current building codes. Inspectors are alert for safety hazards but seldom require upgrading to current codes.
If work is obviously being done elsewhere on the property without a permit, they will probably flag that.
Get a permit for the new work. A furnace improperly installed without a permit could possibly invalidate your fire insurance. If you were to sell your house you would have to disclose that the furnace was installed without a permit.
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I don't think you can get a permit for something which has already been close up. If there is electrical and/or plumbing it would difficult for an inspector to determine if those items were done to code. Also, framing etc. Anything behind the wall probably would not be permitted after the fact.
If it appears dangerous then it could be called out. I imagine it being built in 1960 and you didn't have an issue when you purchased (if you got a loan) it may not even be noticeable as an addition.