A wi-fi router works at 2.4GHz. Can a wireless signal of 2.4 GHz or any other frequency, of sufficiently high energy, damage the device?
Yes, but it's unlikely. Damage from other frequencies will be limited because typical receivers have a pre-select filter after the antenna to filter out other frequencies. In band signals pass through a Low Noise Amplifier (LNA) directly after the antenna, which can saturate if the signal level is high, and fail if the signal level exceeds its maximum input power spec. The most common approach is to put a limiter in front of the LNA to handle any signals that may damage the LNA. However this degrades the Noise Figure of the receiver by the limiter's insertion loss. Another way to prevent overloading is to detect the input power using a power detector and routing it past the sensitive electronics if it exceeds some threshold.
Short answer: yes, any signal can damage the device if sufficiently high energy. However, wifi routers are not usually operating in environments where high power survivability is desired - the focus is more on range and sensitivity. Other sensors such as military radars are designed with robust power survivability specs to deal with jammers, self-reflections, and other electronic countermeasures.
of course. Ronald Reagan's "Star Wars" missile defense is designed to damage the electronic navigation systems on board these missiles. The electromagnetic pulse from a nuclear explosion will certainly deliver enough energy to damage much of the electronic devices within its radius. Satellites, which are transceivers, can be damaged by solar flares and coronal mass ejections.