Replace the CR2032 coin cell battery and see if it helps.  

This requires disassembly, if you are up to it.

Dell Latitude E7240 Owner's Manual

Thank you for the suggestion.

I solved it. Moved the RAM from the current slot to the empty slot next to it. Lo and behold, the laptop has rebooted just fine.

Spoke too soon.

Once the laptop rebooted, I closed the back plate, put the battery back in and restarted the computer normally. Same problem again.

So opened the back plate again, moved the RAM into the original slot again, and restarted the computer without the battery. Computer rebooted as normal. Scared now to turn it off.

Any ideas as to what could be happening?

I would suspect a problem with the memory.  Either dirty connector or internal memory problem.

First thing. Remove the memory module and with a pencil eraser very lightly clean the connector "pins" on both side of the memory module **Do not touch the connector end with your fingers**.  If this does not help consider replacing memory.

I dont think its the RAM module unfortunately. I bought a new RAM stick and the problem is persisting. 

When the RAM module is swapped from one slot to another, the laptop reboots normally. Then if I restart the laptop, nothing happens with just the hard disk drive led just blinking. The only way to get it to reboot again normally is to move the RAM module into the other slot and restart again.

While the laptop rebooted normally, I did run the Dell diagnostics. Nothing stood out except for the health of the primary battery. It was at 48%. When I looked at the event log, it had the following entry listed a couple of times:

"Fail PSA V4228 ERROR 2000:0132. Battery- the battery is reaching the end of its usable life" 

But when I removed the battery, and run the laptop just on AC power, the original problem still persists. So I'm not convinced that it is entirely due to the primary battery.

Is the next step replacing the CMOS battery as suggested? Or there is anything else I can do?