Is using an electric mosquito killer permissible?

In the situation described in the question, using an electric mosquito-killing machine for protection from mosquitoes and flies is permissible according to Shariah. Such a machine contains an electric current, and when mosquitoes or flies come near it, they are killed by the electric shock.

This machine does not contain fire; rather, it operates through an electric current. Therefore, the death of mosquitoes and other insects through this device does not fall under the ruling of killing an animal by burning it in fire. Hence, there is no Shar‘i objection to using such an electric device in mosques, offices, homes, and other places for protection from mosquito bites and the harm caused by them.

The Ahadith mention a prohibition against punishing living creatures through fire. However, the aforementioned electric device does not kill them by means of fire. Therefore, the prohibition mentioned in the Hadith does not apply to this situation.

As for burning other harmful animals with fire, there are details regarding its ruling. It cannot be declared absolutely prohibited or absolutely permissible; rather, the ruling depends on the particular circumstances of each case.

References:

Mishkat al-Masabih by al-Tibrizi (2/306), Chapter: Killing the People of Apostasy and Rebels by Burning:

وعن عبد الرحمن بن عبد الله عن أبيه قال : كنا مع رسول الله صلى الله عليه و سلم في سفر فانطلق لحاجته فرأينا حمرة معها فرخان فأخذنا فرخيها فجاءت الحمرة فجعلت تفرش فجاء النبي صلى الله عليه و سلم فقال : " من فجع هذه بولدها ؟ ردوا ولدها إليها " . ورأى قرية نمل قد حرقناها قال : " من حرق هذه ؟ " فقلنا : نحن قال : " إنه لا ينبغي أن يعذب بالنار إلا رب النار " . رواه أبو داود

Sahih al-Bukhari (3/1098):

حدثنا قتيبة بن سعيد حدثنا الليث عن بكير عن سليمان بن يسار عن أبي هريرة رضي الله عنه أنه قال :بعثنا رسول الله صلى الله عليه و سلم في بعث فقال ( إن وجدتم فلانا وفلانا فأحرقوهما بالنار ) . ثم قال رسول الله صلى الله عليه و سلم حين أردنا الخروج ( إني أمرتكم أن تحرقوا فلانا وفلانا وإن النار لا يعذب بها إلا الله فإن وجدتموهما فاقتلوهما )