If you do the math, you'll see that your worries are misplaced. Because you already would have done it if you knew how, I've done it for you below.
I do think that plastering the world with solar panels would be a real problem, and the logic of living systems suggests that it's a problem we'll have to contend with at some point. There's nothing that inherently limits human energy usage to anything like current human energy usage, so if solar panels are cheap, eventually someone will want to cover the oceans and forests with them.
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However, that will probably not become a problem for more than 50 years. Right now we're only talking about replacing current human energy usage, which is only about 18 terawatts, last I checked, including non-electrical energy. (See, e.g., https://en.wikipedia.org/wiki/World_energy_supply_and_consum...: 162494 terawatt hours in 02017 = 18.537 TW.) With cheap 16%-efficient solar panels and a nominal solar constant of 1000 W/m², that would nominally be about 120 000 km² of solar panels, half the size of Idaho.
But we have to take into account capacity factors, which range from 10% in extremely polar countries like Germany and the Netherlands, through 29% in California, to even higher in deserts. (I'd be very pleased to have some concrete, trustworthy figures on the capacity factors of real utility-scale PV plants in places like Abu Dhabi or Chile.) So we're talking about 400 000 to 1.2 million km², almost half the size of Kazakhstan. Once you set the panels apart so you can angle them toward the equator without them shadowing each other, we're talking about roughly the entire size of Kazakhstan. But presumably Kazakhstan itself is sunnier than that, so a better intuition pump would be the northernmost 20% of Siberia, the part where the permafrost is melting due to climate change, or all of Alaska. (Siberia is 13.1 million km².)
But I proposed building ten times as much solar panel capacity in cloudy places, not three times as much. And that's because, although the capacity of utility-scale PV farms in places like the Netherlands averages 10% year-round, it's only about 2% in the winter, because it gets cloudy. So, if we were talking about a worst case conservative limit in which the whole world is as bad for PV as the Netherlands, and also failed to store a summer harvest of tasty methane to burn in the winter, we need 6 million km² of solar panels, probably spread over 12 million km², the size of all of Siberia.
If 1 m² of solar panel modules weighs 40 kg (I'm too lazy to look this up right now but it's the right order of magnitude due to the glass and aluminum, even though the actual silicon cells are under 300 grams) we're talking about 240 billion tonnes of solar panels, not just a measly few million.
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So, wait, isn't that a huge problem? Doesn't that end up with a country-sized landfill and plastering the world? At 2.4 g/cc and a typical 10 m landfill depth we're talking about 10000 km², which would be, yes, the size of a small country; Monaco is 2.02 km², Cyprus is 10452 km², and Kuwait is 17818 km².
But probably you'd dig the landfill deeper if you were building such a big one. Or pile it higher and just build an earth berm around it instead of digging. At 164 m deep it would be 606 km², the area of Chicago. Technically Chicago is still the size of a small country because there are about 15 countries smaller than Chicago but I think "small-country-sized" is a misleading description of Chicago. I think "less than half the size of Anson County, Georgia, population 22055" is a more illuminating description of 600 km² than "small-country-sized" or even "Chicago-sized". I hope this doesn't offend the inhabitants of the proud sovereign nation of Palau (land area 459 km²).
But this calculation is under the ridiculously pessimistic assumption that we have foolishly located all of the world's solar panels in places like Siberia, Patagonia, or Sweden, because everybody has moved there, and also that they aren't storing up summer methane for the winter. If we assume more optimistically that the solar panels are located, on average, somewhere like California (29% year-round average capacity factor!) and that the people are smart enough to store up methane for the winter, instead of 1.2 million km², it's 400,000 km². So at 200 m deep, 40 kg/m², and 2.4 g/cc, we're talking about 33 km² to bury the planet's 16 billion tonnes of solar panels (6.7 km³).
There are, technically, countries smaller than that: Tuvalu, Nauru, Monaco, and the Holy See. But there's also a pond larger than that in Kennebec County, Maine, called Great Pond.
Great Pond isn't deep enough for all the solar panels, though, because it isn't 200 meters deep. However, the reservoir of Nagarjuna Sagar Dam in Andhra Pradesh holds 8.8 km³ of water (312 TMC or thousand million cubic feet), one of many reservoirs and lakes around the world that are each individually large enough to dump this quantity of solar panels into.
Most of those 40 kg, however, consists of glass and aluminum, the most thoroughly recycled materials in the world, so I wouldn't worry about the landfill. Also, there are some 50-year-old solar panels already, and they mostly still work, just at reduced power output.
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But what about "plastering the world"? Doesn't a million km² of solar panels amount to "plastering the world"? No, the world is 510 million km², so we're talking about plastering 0.2% or 0.1% of the world.
But cheap solar panels mean cheap energy, which means human energy consumption can expand dramatically. Historical societies that had high human development, like Classical Greece and Rome, only had it for a small upper class whose wealth was underpinned by the forced labor of slaves. Right now we use 2300 watts per person (18 TW ÷ 7.7 billion people) which is equivalent to about 23 "energy slaves" per person. The solar resource is 127500 terawatts, large enough that by plastering the world with solar panels we could have 1000 times that, with the equivalent labor of 23000 slaves at the disposal of each person. The temptation to do this, despite the attendant destruction of the biosphere and the alternative of space-based solar power, will be strong. But that's more than 50 years in the future.
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Let's consider, as an example, just the Netherlands, and disregard the wind energy they've been famous for exploiting for centuries.
The Netherlands uses about 100 GW (900 TWh/year), of which about 120 TWh/year is electrical (https://en.wikipedia.org/wiki/Energy_in_the_Netherlands). It's 41865 km² with a 10% countrywide capacity factor for utility-scale solar, dipping to about 2% during the winter months; presumably this would get worse if you started having to build your solar farms in random places instead of the sunniest places in the country. Let's say, pessimistically, 6% and 1.5%. 41865 km² at the solar constant of 1000W/m² is 41.9 TW, about twice world marketed energy consumption. At this pessimistic 6% capacity factor 41865 km² of mainstream 21% efficient panels would produce 530 GW, 5 times the country's current energy consumption.
This means that by covering 20% of the country in solar panels you can supply its whole energy usage with 6%-capacity-factor 21%-efficient solar farms. At today's high €0.33/Wp prices the required 1700 GWp of solar panels would cost €570 billion, 28 weeks of the Netherlands' GDP of US$1055 trillion/year. Installation and balance of plant (inverters, etc.) would cost (guesing) another €700 billion. And then you need storage, which is another significant but smaller cost. These costs will almost certainly go down in coming years.
You'd have to store methane for the winter, though. Or use wind.
This 1700 GWp is 8100 km² and at 40 kg/m² would be 320 million tonnes of panels. At 2.4 g/cc this is 0.13 km³ or 1.3 km² of 100-meter-deep landfill. Hopefully you can imagine that 1.3 km² of landfill would not be a major catastrophe for the Netherlands.
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Basically your concern is like a kid worrying that if his parents buy him a lollipop they won't be able to afford this month's rent. It's not completely disconnected from reality but it's way out of proportion.