Electronic device with heat dissipating electromagnetic interference shielding structures

An electronic device may have a metal electromagnetic interference shielding enclosure. The enclosure may have a bottom wall, vertical sidewalls that extend upwards from the bottom wall, and a lid that covers the enclosure to define an interior cavity. Power supply components and other electrical components may be mounted within the interior cavity. A printed circuit board on which integrated circuits and other components are mounted may have an upper surface that faces the bottom wall of the enclosure and an opposing lower surface that faces a metal plate. Fence structures may be used to help shield components mounted on the printed circuit. Heat may be dissipated from components on the printed circuit into the bottom wall and into the metal plate. A plastic housing may be used to house the shielding enclosure, printed circuit board, components mounted on the printed circuit board, and the metal plate.