Why is my ps4 constantly ejecting and beeping?

I encountered that problem with the slim and 1st gen PS4 consoles. Both had a faulty eject button that started making contact without pressing and would eject anything that was inserted in tray. Also, the slim would make a periodic eject sound with nothing in tray. I was able to replace the switch for the slim from another console and fixed the problem. Did the same thing with 1st gen console and fixed it as well. However, since the 1st gen has daughterboard married to motherboard then the game disc could not be played. As the eject button is fixed onto the daughterboard/logic board of 1st gen consoles. But since you said you removed eject button and did not solve problem somehow motherboard could be bad? I am not sure why yours is still acting up even with eject button removed. Try cleaning the inside of the optical drive, final resort.