This is a fascinating topic because there is no practical way of knowing what food-related items (containers, utensils, washing up liquids) are safe to heat and what aren't. Indeed no one can track the makeup of everything in their kitchen that comes in contact with food.
Then there's the topic of items that are safe to use only in ideal conditions. Like Teflon in cookware that is said to be very safe unless it overheats, chips, or degrades in some way.
The more I think about this, the more it sounds like a blind spot for most people. So I'd like to know whether heated-up washing-up detergent/dishwasher detergent is associated with something unhealthy in a proven (or reasonable) way.
The plastics industry dodged the issue/controversy by eliminating BPA. While BPA is by far the worst endocrine disruptor, it is extremely likely that all the plastics that you are currently "safely" using contain endocrine disruptors.
You should be avoiding storing consumables in plastics as much as possible. At that point microwaves become perfectly safe.
