Franz Böhm was born and raised in Gerlingen, northwest of Stuttgart. At 16, he made his first short film, Harmony of Others, about interactions among teenagers from different social backgrounds. In 2017, he created the medium-length, fictionalized documentary Christmas Wishes, depicting the lives of two homeless people in Berlin, for which he lived with the homeless for a while. This project opened the BW Youth Film Prize in 2017 and won several awards.

Böhm's international debut came in 2019 with the short film Good Luck, which premiered at the British Independent Film Festival and won the "Best Cinematography" award.

His feature film debut and international breakthrough occurred in 2021 with the documentary Dear Future Children. This film won the Audience Award at its world premiere at the Max-Ophüls-Prize Film Festival, at the Hot Docs International Film Festival, and at its Swiss premiere at the International Film Festival and Forum on Human Rights. The 89-minute documentary, focusing on young international activism through the experiences of three activists from Hong Kong, Uganda, and Chile, was nominated for major international documentary film festivals, including CPH:DOX.

In 2022, Böhm became the youngest director to win both the Audience Award and the Main Prize at the German Documentary Film Awards.