Assuming the system still operates, you should cease using it until the problem is fixed, as it is a potential safety hazard. Most likely the fix is to replace the system board.
I know this is an old thread, but I searched on "inspiron smoking" and got lots of hits around the top case melting. This thread sounds like my issue. My Inspiron 15 was manufactured in September 2013. For it to fail this soon, and in such a spectacular manner bothers me.
This past weekend, the screen went dark while I was browsing the web. I tried pushing the power button, unplugging it and plugging it back in, etc. I flipped it over and removed the battery (which was functioning fine, btw). Upon reinstalling the battery smoke came pouring out of the vents on the bottom of it. I frantically removed the battery (I've seen LiPo batteries explode first-hand). Just for fun, I put the battery back in yesterday to see if maybe it had magically repaired itself and smoke immediately came pouring out of it again.
At first I was willing to accept the fact that it's dead and move on. But I got to thinking - what if this had happened when I wasn't home? I store the thing in a drawer when it's not in use - it could have set my house on fire!
If anyone at Dell wants to examine it let me know - I don't know what to do with it.