The generally accepted medical wisdom seems to be that sweat is an insignificant mechanism for toxin elimination. However, there have been studies that suggest that wisdom may not be entirely correct.
For example, Genuis et al. found that some toxic elements were preferentially excreted in sweat, including elements that were not found in blood serum. That suggests those elements are bound in tissues and therefore not readily removed by the liver and kidneys.
They conclude that:
Sweat analysis should be considered as an additional method for
monitoring bioaccumulation of toxic elements in humans.
Note that they do not conclude that sweating plays a significant role in toxin elimination.
Note also that the sample size in this study was very small, and the author has numerous publications based on this single study.
A literature review of the subject found support for the idea that certain toxins are excreted in sweat, sometimes in higher concentrations than urine. They also noted as Genuis did that concentrations of some elements can be found in sweat that aren't detected in plasma.
In individuals with higher exposure or body burden, sweat generally
exceeded plasma or urine concentrations, and dermal could match or
surpass urinary daily excretion. Arsenic dermal excretion was
severalfold higher in arsenic-exposed individuals than in unexposed
controls. Cadmium was more concentrated in sweat than in blood plasma.
What I cannot find is research showing that sweating produces any clinically significant benefit. Just because a compound can be found in sweat does not necessarily mean that sweating more will have a beneficial effect, and in fact no benefit to excessive sweating is known while there are known negative consequences, including potentially lethal ones.
I conclude from my research that the question can't be answered at this time, but there is sufficient evidence to justify additional research. I suspect that the most likely outcome will be new and/or improved testing methods rather than treatment methods.