Is it permissible to kill household insects like ants and cockroaches by burning them? And if not, what should we do?

Harmful insects are to be killed with insecticides, not by fire. It is mentioned in the Hadith: "There are five animals, all of which are wicked, and can be killed in the Haram (sanctuary) and outside it: the mouse, the scorpion, the crow, the kite, and the rabid dog" (Muslim 2071), and "the snake" was added. These wicked animals and other harmful creatures like ants and cockroaches should be killed with insecticides.