I bought my first diesel new off the lot in '96, so it's taken me almost 13 years to accumulate enough knowledge to keep mine going.
I'd say it's darn near impossible to list the common things that go wrong. From common suspension bushings, to sensors on the computer controlled motor, to door pins, serpentine pulleys, fuel pumps, injectors, rusted engine oil pan and rear diff covers... The list is almost endless, but same goes for about any used vehicle.

Highlighted above, I would not categorize these trucks as 'Cheap to operate'. When, not if, but when repairs need to be made they are expensive. Example, gasket set alone for the motor runs around $600. You can buy same thing for 350 Chebby for like $50.
You have to really need a Power Stroke to justify it's expense. 4 gallon oil changes, fuel costs almost twice what gasoline costs, etc. I went almost 6 years driving a gasser ('96 was parked, needed a work truck and wasn't going to abuse my 'new' truck) cuz the maintenance was too much.

BUT, this depends 100% on previous maintenance. I had a '95 with 160k miles on it and the motor was trashed tired. I have a '97 with 250k miles and runs like it's new. Maintenance, maintenance, maitenance...

Not trying to scare you out of it, but I've had 6 OBS Power Strokes and I know what it takes to keep them going. They are great if taken care of, but I don't really need one, I just want one, and I pay the price in maintenance.
I went back to diesels when I started having to plow our jobsites in the winter time. Gas truck was getting 7 mpg, F-350 diesel got 14. Unless you work the truck I doubt you'll see much, if any, fuel savings. Towing or hauling the Power Stroke really shine with their power and economy vs gassers.

Good luck with what you decide.