I believe it’s especially prevalent among owners of Japanese cars- the owner replaces the stock wheels with larger, sometimes to the point where they have to have their wheel wells altered to accommodate them. I think it’s mostly for aesthetics but handling is affected also. Do the stock wheels actually have an advantage in any area? I mean, aside from tires being cheaper 
Increasing wheel size is equivalent to increasing gear ratios, so smaller wheels make the car accelerate faster. Of course, there’s an offsetting penalty in top speed, assuming the car isn’t aerodynamically limited. You can change the gearing to compensate for larger wheels, but most tuners don’t, AFAIK.
If you’re talking about width, narrower tires have less rolling resistance.
A freind of mine, many years ago, bought a rattletrap old truck and we set off on a road trip.
After he used a tube and a half of grease on the grease points of the frame… should have been the first clue. The second clue was that we could see daylight through the bottom of the rusted doors.
After about 90 miles of cruising 70mph we stopped, I checked the map distance and time and concluded we’d been doing 55mph. I think the previous owner cheaped out for smaller tires.
Look at most cop cars. They don’t have fancy alloy wheels. They have steel wheels. One reason is they are actually stronger and take a lot more abuse such as potholes and curbs.
Bigger wheels may mean a lot of things. You can have a bigger wheel, but have the same diameter and width of the tire. In that case the only reason besides looks is you can install larger brakes and the brakes may cool better.
Wider tires may give better cornering, but at the expense of gas mileage. Their can be problems with fitting inside the wheel well especially on the front.
Taller tires might give better gas mileage especially on highway driving. Either taller or smaller diameter tires than stock will mess up the odometer and speedometer reading unless adjusted.
Taller tires can also have problems fitting in wheelwells.
If you get taller, wider tires fitting in some wheelwells can be a real issue. Back in the day you’d often see wider tires on the back than the front as the rear wheels just point straight ahead.
I believe the OP is talking about larger wheels with lower profile tires, yielding the same effective tire diameter, so the effect on acceleration would be nil.
(ETA: on rereading, maybe that’s not what s/he is talking about, but most of the time when I see non-stock wheels on cars, it’s to use lower-profile/wider tires)
Several years ago, Car & Driver did a comparison test (not online, AFAICT) on several cars with the same tires on different size wheels (and obviously varying aspect ratios). I don’t remember the exact results, but there is a law of diminishing returns regarding the expense of lower profile tires/larger diameter wheels. I think they found the sweet spot on most cars to be 17-18" rims.
The disadvantages of larger tires is (other than expense) the increase in unsprung weight and reduced ride quality.
There’s also this thread here.
Found the article I was talking about; it’s here. Apparently they only tested on one car, a VW Golf.
There are two different things here.
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You can have a larger wheel with a correspondingly larger tire, which may require you to increase the size of the wheel well also. You lose low-end torque on initial acceleration due to the gearing differences mentioned upthread. You also gain traction from the larger tire (and its correspondingly larger footprint on the road) at a cost of increased friction and correspondingly decreasing gas mileage. On my pickup truck, larger tires also increased the stability of the truck at higher speeds (note - you also have to adjust your speedometer or you will indicate a lower speed than you are actually driving).
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You can have a larger wheel and a much thinner low profile tire to end up with the same overall outside tire diameter. This was done originally for sports cars, since the smaller tire sidewall doesn’t flex as much and allows for tighter gripping of the tires in high performance turns. Then everybody figured that hey these are cool new sporty tires so they must be good for everyone who wants to be cool! And the new look became popular. Then those folks quickly learned that low profile tires don’t cushion the road bumps as much, resulting in a much harsher and less comfortable ride. Most people also don’t get any real benefit out of them since the performance differences between low profile tires and regular tires doesn’t show up until you start taking turns well in excess of the posted speed limit.
I don’t know if this practice in continuing, but a few years ago new car dealers would note in a vehicle’s permanent service record owner-installed aftermarket wheel. Their concern stemmed from replacing stock rims/tires (which are designed for the vehicle’s suspension, weight, brakes, center of gravity, etc.) with absurdly large rims with very low aspect ratio tires. A prime example is an SUV/Pick Up that may have come with 17" rims and 75 series tires on which the owner put 24" rims with a 20 series tires. Not only is the weight of the rim excessive, there is very little absorption/flex in the tire. Either can have a very dangerous effect on handling and braking. The vehicle manufacturers asked the dealers to note these installation fearing possible accidents and lawsuits.
well, it depends. stock wheels go through far more stringent validation testing to make sure they will live for 150,000 miles without cracking, or corroding, or the finish peeling off. Aftermarket wheels have no such constraints.
now, if you’re asking about “plus 1” or “plus 2” wheel sizes, where the wheel diameter is increased but combined with a lower profile tire to maintain the same rolling diameter, that can be done to improve handling. The deflection of the tire sidewall can make your vehicle’s handling “squishy” or "squirmy.’ I’ve seen pictures of '70s and '80s cars undergoing handling tests, and the tires were so far over on their sidewalls that I’m shocked they didn’t bust off of the bead and deflate.
finally, if you’re talking about people who put giant wheels with rubber-band tires on their cars, necessitating that they either lift the suspension or cut the wheelwells for clearance, well, those people are just assholes.
Amen.