Note: my understanding of chemistry involving molecules with more than a few atoms is severely limited.
An answer to your question requires understanding of concepts in physical chemistry and biochemistry, and careful use of technical terms such as "metabolize" and "heat". Thus, this answer might be a bit frustrating today, but maybe helpful to motivate further study.
I can imagine one main disadvantage of a process using sunlight to metabolize sugars would be that wherever there is little/no sun (nighttime), plants don't grow.
When we say "metabolize a substance", usually we mean starting with that substance and turning it into something else. The process that makes use of sunlight results in the biosynthesis of sugars. The plant (and this is something you would usually not say) is "metabolizing water and carbon dioxide to produce dioxygen and sugar".
Over their lifespan, photosynthetic plants utilize light to grow. However, they make sugars during the day and store them (and distribute them to non-photosynthetic parts of the plant, e.g. the roots). The plant grows just fine in the absence of light as long as there is some light every now and then. So leaves can grow at night, and roots can grow in the absence of sun exposure. In fact, seeds grow to plants using the nutrients (usually mostly starch or fat) stored in the seeds combined with oxygen in air while photosynthesis is not yet possible.
So plants can grow without light, but photosynthetic plants rely on light to capture most of the (Gibbs) energy that they expend.
Heat is more abundant than sunlight and could theoretically be an alternative (think units buried deep underground near geothermally active areas).
"Heat" is a technical term reserved for the transfer of thermal energy. From the context of the question, I would think you mean "thermal energy".
Sunlight, wind, and electricity are more "useful" sources of energy than thermal energy. You can turn sunlight, wind, and electricity into thermal energy, but you can't "extract" thermal energy from a sample to make e.g. electricity. What is possible is to make e.g. electricity using a hot and a cold sample, where the cold sample gets heated up. You see that in coal, gas or oil-fired electrical power stations, that often are next to rivers to have access to cooling water.
How is this related to making sugar and dioxygen from carbon dioxide and water? This direction of the reaction goes away from equilibrium, and you have to do work to force it in this direction. Work is an umbrella term for electrical work, photochemistry, or mechanical work. (The same is true for making heat flow from a cold to a hot sample, like a refrigerator.)
Is there a heat-based (artificial or natural) replacement for photosynthetic processes?
Yes, you could use a cold and a hot reservoir to make some electricity (for example with a turbine), use that to electrolyze water to dioxygen and hydrogen, and let the hydrogen react with the carbon dioxide to make molecules like formaldehyde or methanol or methane. From there, you still have to make sugar. There are probably some organism that are able to do that in the absence of sunlight.
What you can't do is cool down a hot substance, and use the energy to turn carbon dioxide and water into sugar and dioxygen.
[from the comments] There are devices such as photothermovoltaic cells or thermoradiative cells that can input heat and output electricity.
Yes, these hot-of-the-press devices are pretty cool (pun intended).

The radiative heat part would not work if the system were at thermal equilibrium. So there is still heat transferred from a hot to a cold part. When they calculate the efficiency, they get a familiar result.

So while there are some technical advantages of this over a Carnot cycle, the second law still governs this technology.