Also, it seems that it will stay on for a while when it has not been running for 30 -45 minutes. In that case I can boot up, log on to windows, run a website for a few minutes and then crash.

But immediately after that crash, I have to do a hard shut down to power off. I wait 2 minutes or so until the 3 4 LED lights are off. Then I reboot and it may boot up, but sometimes then it goes right into the beeping and 3 4 LED error lights again. So I think there is something related to the time it has been running? 

First off, Dell has not tested this model for compatibility with Win 10 and offers no drivers for that OS. Did you find/install all the right Win 10 hardware drivers, including chipsets, video, etc. yourself?

According to the 780 service manual, beep codes for this system come in groups of three. So you may not be counting correctly. If you're actually getting 3-3-1, that's: NVRAM power loss. Codes between 1-3-1 through 2-4-4 are RAM errors.

LEDs 3,4 is a RAM error. Do you know the new RAM is compatible? Dell PCs can be very fussy about RAM even when the specs seem correct.

I'd replace the motherboard battery to start because that could be related to possible NVRAM power loss, assuming the beep code is 3-3-1. It's a 3-volt CR2032 lithium ion battery, ~$2 at discount stores. Even if that doesn't solve the problem, at least you'll have ruled the battery out.

Since both new and old RAM give same 3,4 LED code, if it were me, I'd go back to trying the old RAM at least until you resolve the problems because the new RAM may or may not be compatible.

Remove old battery, re-install old RAM, then press/hold power button for ~30 sec. Then install the fresh battery. See if it boots and runs correctly now.

If that fails, remove the RAM module from slot 2, leaving only the one in slot 1. Remove battery and press/hold power button for ~30 sec, reinstall battery and see if it boots now. Then swap the other old module into slot 1 to see if you can ID a bad module. If neither RAM works in slot 1, try using the other slot, and do the "battery thing" each time you change the hardware. Maybe you can ID a bum motherboard slot.

If you can't get it to run correctly with only 1 old RAM module in either slot, remove all RAM modules and do the battery thing again. Now try to boot without any RAM. You should get LED 1 error code meaning: No RAM modules detected.

Are all the fans working? You might have a heat problem.

Make sure fans are clean and turn freely Use canned air to blow out all the dust bunnies from fans, heat sink, motherboard, and air vents.

Possible the thermal paste has dried out. In this case, you'll have to remove the heat sink, clean both surfaces and apply a thin coat of new paste (eg, Arctic Silver) and then reattach the heat sink. Check manual for instructions...

"First off, Dell has not tested this model for compatibility with Win 10 and offers no drivers for that OS. Did you find/install all the right Win 10 hardware drivers, including chipsets, video, etc. yourself?"

Just in my experience.  Did a fresh Windows 7 install on an OptiPlex 780 yesterday using Dell Windows 7 disc.  Then upgraded to Windows 10 right away.  All my hardware drivers were found immediately .  Windows 10 running very smoothly on this older computer (6GB RAM). 

Thanks Sprung. I will remember that. Last week, when it was running, it seemed to run very well on 10 for two days. I am thinking it is hardware. For now, I need to figure out how to turn it on...

Rohe - I did not install anything for Win10 aside from what was on the installation stick.  It seemed to run fine for two days last weekend.  Used IE instead of chrome and was able to run the full 45 min Dell Diagnostics and everything checked out. The history is under my service tag above (the tray out failure was due to it being too close to the desk and hitting that.) 

I will try the motherboard battery and RAM see if that helps.  I double checked with two different sites about compatibility of my new RAM with the motherboard. 

The fan is running fine and it is clean inside and out. I will also look at the heatsink when I am in there.  

RoHe - I replaced the CMOS battery and that seemed to do the trick.  However when I shut it all the way off and turn it back on it says "Voltage is low" press F1 to start, F2, F3 etc.  I hit F1 and it runs fine now.  Not sure if this is going to cause me problems down the road.

Go to the grocery store and buy a fresh Duracell DL2032 or Everedy CR2032 battery. Low voltage means the battery is already 9 years old or was in a system that had no power for months.

They are usually in the front or in the back by the *** because blood sugar machines and around the neck hearing aids use them.

The ones marked medical device work fine in your PC.  They last 10 years in a cool dry place when not installed.

Does sound like the new battery may not be fresh.

Be sure to hold the battery by its edges when installing and avoid fingerprints which can affect contact with the socket.

And make sure the battery is fully and correctly seated in the socket.