Thanks for the endorsement! And the homework assignment.
- While others surely have me beat for years of experience, I've owned my avatar car for 25 years, a not-published-here 65 SF for 30 years, as well as several others that are long gone. I was learning about Furys before the internet existed and did my first disc brake swap in 1993 or so, by looking at Napa suspension catalogs to figure out what would fit. I also know enough to realize I don't know it all. With that said...
Lots of folks have covered lots of good things already.
#1 If you have some old car experience, there isn't really much on a 67 Fury that is 'secret' regarding an inspection, and many repair parts can still be bought new, some can't, but for those we have work-arounds.
#2 If not experienced, take a car-savvy friend. Even if you are - take a friend. A 2nd set of eyes can help if you get smitten and overlook faults.
#3 Wear 'dirty' clothes and take rags/gloves, and slide underneath to check for rust. Trunk floor, bottoms of quarterpanels, floorpans, framerails. Framerail rust is generally confined to the horizontal sections of the frame (the parts you can see). Meaning - I've never seen a C-body with good horizontal sections that was then surprise-rusted above the rear axle, or up near the front suspension. Same thing with sheetmetal - you won't find good floorpans but with a rusty driveshaft tunnel.
I'm a bigger fan of the 68 than the 67 - I like the grille better on the 68.
BUT - From what I see in that one pic, I would like this prospect car enough to consider buying if it checked out (meaning, you have found an eye-pleasing example)
Like most Mopar cars from the 60s, most of the aluminum trim, grilles, bumpers, taillight lenses, and 1/2 of the sheetmetal is specific to that car, and that year. Some sheetmetal parts can interchange, but usually only for 2 years at most. If this car is complete and in GC that's all that matters.
Being a Sport Fury brings bucket seats, possibly console, and the H-code 383-4 barrel is a nice upgrade over the G-code 383-2 barrel. It is common for folks to upgrade and call it a 4-barrel car, so check the VIN - 5th digit is the engine. In 67 the H-code got better exhaust manifolds than prior-year 4-barrel engines, they curve upward a bit, while the standard ones run parallel to teh valve covers.
An H-code 383 doesn't make this car rare or super-special, but it is the 2nd most powerful (other than the rare 440) and it will make for a well-balanced car that will be fun to drive. You won't win races with it, but anybody that rides in it with you will recognize it's got some bravado.
Check the firewall for rust, from near the heater hoses, across to the wiper motor, and master cylinder. Look for pinholes. These cars can get leaves and pine needles down in the cowl, thru the vents at the windshield base, and the debris can hold moisture. This car looks well-loved, but check it anyway. There could be issues on the dashboard-side of the cowl (harder to inspect) but if the firewall is OK the interior side likely is also.
This car has the 14" 'mag' wheelcovers, so it has drum brakes (unless someone converted to SSBC brakes). That means you've dodged the 66-68 Budd disc brakes, which are pricey to repair. Flipside - you may want to convert this to discs at some point, and that can be a $1000+ proposition depending on which parts you elect to use. (but it's a fabulous upgrade for safety)
Red circle - check here for rust, and also between the trim and back glass. These would be the first spots for rust to appear. Look for any bumps under the vinyl. In my experience, this particular roofline (which it shares with the 66 300) isn't as bad for rust as some of the other C-body models.
Other red circle - someone else mentioned it - this area needs inspection.
Blue circle - might be reflections, might be peeling clearcoat if it had been repainted that way. You'll know it when you see it.
Green - this looks very much like someone wiped it dusty. Whole car looks dusty. No big deal there, though.