It's obvious that now, there is always an electric light 'burning' on earth somewhere. How far back do we have to go until that was not the case?
I don't think we can get an exact answer, but we can at least set some parameters and make a reasonable guess.
First, This light bulb in Livermore, California has been burning since 1901. It's been off for a few brief periods since then, but the point is to demonstrate that early filament bulbs lasted a long time and were left on.
The first electric arc street lighting was installed in Los Angeles in 1876 and in London and Paris in 1878. These were likely turned off during the day. Or rather, the electrodes burned out quickly and were not replaced until needed again.
Now, we go back to the time just after Edison perfected his incandescent light bulb. The Menlo Park Museum website has this to say:
Finally, on October 21, 1879, Edison’s light bulb burned for a continuous thirteen and a half hours. The following bulbs lasted for 40 hours and Edison and his team worked hard to light the laboratory and his home with several of the new light bulbs for Christmas. On New Year’s Eve of the same year, Christie Street became the world’s first street to be lit by incandescent light bulbs with the help of a power system designed by Edison.
It's unlikely that the lights on Christie Street stayed on all the time; it's a bit more likely that the lights in the Menlo Park lab were always on. In 1880 Moseley street in Newcastle was lit by J.W. Swan's independently-developed incandescent lamps.
Edison and Swan soon began selling lighting systems to early adopters, wealthy individuals who lit their mansions as status symbols. So the chances of some light somewhere being on all the time increased.
And then came the summer of 1882:
In the summer of 1882 [Edison] began setting up a large generator plant at Pearl Street in downtown New York City. He fitted all the office buildings and homes on Pearl Street with about four hundred of his incandescent bulbs. On September 4, 1882, hundreds of people gathered on Pearl Street to witness a never before seen spectacle; at 3 p.m. the generator was turned on and the street was lit with electricity.
My reasoned guess is that this moment in 1882 is your answer. Although an 1890 fire put the Pearl Street generating station out of commission for 11 days, it is likely that one of the buildings on Pearl Street had at least one light bulb on the whole time it was in service. In the meantime, of course, the use of electric lighting spread rapidly (in 1888, Tamworth NSW was the first town in Australia with electric street lighting), reducing the chance of a completely electrically-unilluminated Earth as time passed, until the Livermore bulb first turned on in 1901.
The first electric lighting device was not an electric bulb but the "Volta's arc", also known as "electric arc", see Wikipedia. In 1801 Humphrey Davy described an artificially produced electric discharge in the form of a spark. The first continuous arc was achieved by Vasili Petrov in 1802. It was actually used as a light source before the invention of the bulb. Electric arc is still used as a light source in the powerful search lights, beacons etc.
But of course natural "electric light" always existed in the form of lightning. So the Earth was always illuminated by electric light from time to time.