No chime or display but loud fan

What does the onboard diagnostic LED’s telling you? You’ll find them at the bottom left edge go the logic board often coved with a strip black tape.

LED #1 - Indicates that the trickle voltage from the power supply has been detected by the main logic board. This LED will remain ON whenever the iMac is connected to a working AC power source. The LED will remain on even when the computer has been shut down or put to sleep. The LED will turn off only if the AC power source is disconnected or the power supply is faulty.

LED #2 - Indicates that the main logic board has detected proper power from the power supply when the computer is turned on and that logic board voltage regulators are functioning properly. This LED will be ON when the computer is turned on and the power supply is working correctly.

LED #3 - Indicates that the computer and the video card are communicating. This LED will be ON when the computer is communicating properly with the video card. If LEDs 1 and 2 are ON and you heard the startup sound, but LED 3 is OFF, then the video card might be installed incorrectly or need replacement.

LED #4 - Indicates that the computer and the LCD panel are communicating. This LED will be ON when the computer is turned on and video signal is being generated. If the LED is ON and there is no image on the display, then the LCD panel, the LED backlight board, or the cables between might be installed incorrectly or need replacement.