You would need:
Energy: Nuclear power, fission or fusion. Nuclear waste recycled or glassified and buried.
Electrical Food Chain:
Photosynthesis is just a way to knock electrons off atoms. In principle, it can be done with electricity alone. There's active research on this. Probably soon we'll have microbes that can produce sugar, fats etc from a current.
More simply synthesize high energy compounds and let microbes eat them.
Lastly, convert electricity to light, put plants under lights, stack as high as needed.
Regardless of method, food is produced in highly compact areas in quantity year round.
Carbon cycle remains closed.
Raw Materials No real changes, everything recycled simply because there is no place to put waste. Mines and the like are just in the basement.
The real problem with a planetary city is heat. Urban areas are already heat islands, always hotter than the surrounding natural terrain. Cover the entire planet and you'd heat the entire planet. Might actually cover the entire top in reflectors because really you don't need solar energy (solar panels are thousands of times less dense than nuclear power and they'd generate more waste heat per kilowatt.)
Presumably the planet starts as a terrestrial planet and has seas. Seas are the major heat conveyers on earth. They are powered by solar evaporation which changes density causes changes in density of seawater at top causing it to fall. If you build over the ocean you will stop the heat conveyers and the poles and the equator will have radically different temperatures which will produce storms. Likely, they will dump some waste heat back into the ocean to mimic solar effects and keep the conveyer going.
The real problem will be radiate enough heat into space. There is a phenomena called "heat pollution" that will occur when just the heat dumped into the atmosphere by technology starts to alter climate and drive storms.
Most likely a civilization advanced enough to build a planet city could figure out alternative radiators for the planet e.g. radiating lasers, magnetic radiators or just a bunch space elevator like structures that were just radiators carried huge amounts of heat from the ground to space.
Economics: To my mind, the major technical road block to a planetary city would be one of what pragmatic or economic forces would drive that many people to pay the enormous price of cramming that many people together?
Population will not grow as populations urbanize because children cost a great deal in urban areas and unlike under farming, produce no income until well into adulthood. Long before a planet gets paved over, population growth will be flat or even declining.
Dense urban cores, especially ones with skyscrapers, were justified by the need to cram lots of people close together so they could communicate with analog technology. Such dense cores are already technologically and economically obsolete, though not with out their lifestyle appeal for many.
We've long passed the stage where people had to be in New York to hit the big time. Major corporations are spread all over the world in urban areas big and small. It's as easy to send email around the world as in the same building.
I think the most likely scenario might be a completely non-economic reason to cram hundreds of billions of people on a single planet.
- A prison for a very large galactic civilization, perhaps locking up entire societies for some reason.
- A refuge of some kind, everyone is there because it's the only place to survive. It could be a hiding place or a fortress. If the latter they would be like people fleeing the barbarians to hide in the castle.
- Planet city, but mostly empty. For whatever reason, a lot more city got built than needed. Runaway autonomous construction robots is a scenario I've seen once or twice. In that scenario, the city becomes the geography with 99% of it empty but maintained the robots. Spooky.