Toshiba Satellite L840 boot from external disk without internal HDD

First of all, is there any RAM in the computer? If not, this could be the cause. If there is then you should get a BIOS splash screen at the very least without a hard disk or battery (assuming it's plugged in, of course!)

Though not conclusive, the fact that the fan runs would seem to indicate there is still some life in the old bird, as the fan is firmware controlled. On the other hand no display and not even any beeps isn't a good sign. If it is beeping, google for toshiba satellite beep codes to decode them. (No RAM found often results in a beep code.)

The other thing to try is to take out the motherboard and look for signs of spillage. If anything looks suspicious, clean it up with isopropyl alcohol. And while you're there, look for loose cable connectors, in particular the display cable. I worked on one laptop where it was partially disconnected at the LCD panel end. Oh, and try an external monitor. Does that show a splash screen?