I keep hearing people exercise with clothing that makes you warmer and therefor make you sweat more. However, I have also heard that this only dehydrates you and is not necessarily tight to burning more calories. There is a logic I see where your body burns more calories by trying to keep your temperature low, but I am not sure how healthy it might be or if the burning of calories is high enough to make a difference.
The affect of heat stress on metabolic rate during exercise is unclear, but performance is likely to be impaired.
Chapter 3 of Nutritional Needs in Hot Environments: Applications for Military Personnel in Field Operations1 gives a narrative review of the state of the literature on this question (as of 1993, yeah, it's old, I'll do more research into recent studies). In the section on metabolism, the authors report:
Acute heat stress increases resting metabolic rate (Consolazio et al., 1961, 1963; Dimri et al., 1980), but the effect of heat stress on an individual's metabolic rate for performing a given submaximal exercise task is not so clear (see Table 3-1). Such an effect would influence the calculation of the heat balance and might have implications for the nutritional requirements of individuals exposed to hot environments. Many investigators report that to perform a given submaximal exercise task, the metabolic rate is greater in a hot than temperate environment (Consolazio et al., 1961, 1963; Dimri et al., 1980; Fink et al., 1975). Some investigators, however, report lower metabolic rates in the heat (Brouha et al., 1960; Petersen and Vejby-Christensen, 1973; Williams et al., 1962; Young et al., 1985). Heat acclimation state does not account for whether individuals demonstrate an increased or decreased metabolic rate during submaximal exercise in the heat. However, other mechanisms can explain this discrepancy. Most investigators have only calculated the aerobic metabolic rate during submaximal exercise, ignoring the contribution of anaerobic metabolism to total metabolic rate.
So the findings reported here range from "reduced metabolic rate" to "no change" to "increased metabolic rate".
However, the evidence is more clear that performance is likely to be impaired. In this same narrative review, studies testing VO2 Max under heat stress conditions were examined, and the results are much more homogeneous than those for metabolic rate: maximum oxygen uptake is impaired under high temperature conditions. So if performance in a given workout is important to you, that is, the adaptations you are selecting for in a given training bout are contingent upon your performance during exercise, working in conditions that impair VO2 Max are going to be suboptimal.
1 Institute of Medicine (US) Committee on Military Nutrition Research; Marriott BM, editor. Nutritional Needs in Hot Environments: Applications for Military Personnel in Field Operations. Washington (DC): National Academies Press (US); 1993. 3, Physiological Responses to Exercise in the Heat.
I don't think that your body can burn more calories to keep itself cooler, it's not like an air conditioning system that can actively cool. It can actively heat, itself of course, but that's directly generating heat energy and there isn't a "cold energy" that can be generated by mammals - unlike with the use of refrigerants and electricity!
The first article that came back for me in a Google search is in Women's Health Mag and says:
Think your workout was extra hard because you're drenched in sweat? Tbh I did, but that’s not always the case... sweating does not burn calories. Nor does it burn fat or offer a sign of exercise intensity
You are correct that it can result in a weight reduction through water weight, which is very rarely desirable. One counterexample is when fighters need to "cut weight" before competing - the day before the fight, they must not exceed the maximum weight for their weight division and it's common practice to (quite severely, in many cases) deplete themselves of water so that they can achieve this figure. There are many methods that may be utilised but exercising or sitting in a steam room with layers of clothes can force a lot of water to be lost. Apparently (according to this answer to a Quora question) it's not unusual for a fighter to immediately put 10-15lb back on in water weight between the weigh-in and the fight itself.
Losing water weight does not contribute to fat loss, though, and trying to exercise when dehydrated (or even when too hot) will have a negative effect on your performance.