As far as I can tell, this seems not to be quite a formal, well-defined term. Rather, it's the use of an iconic cultural touchstone as shorthand for issues and tropes that come up often - and, just as the original creation can be interpreted in different ways, can be admired or criticized, people's use of the phrase "Devdas syndrome" seems to depend on their own interpretation of the story.
Devdas is a tragedy -- rich Devdas and middle-class Parvati are in love, but the match is prevented by family, class, and culture. Devdas goes into a self-destructive spiral of melancholy and alcohol. "The Devdas Phenomenon", by Corey K. Creekmur, summarizes:
Three key events carry the story to its hopeless conclusion:
- Devdas writes Paro an insincere letter denying his love for her, which he attempts but fails to prevent from being delivered;
- prior to her wedding, Devdas, breaking a childhood promise never to hit Paro again, scars Paro’s beautiful face (originally with a fishing rod), marking her with a symbol of his enduring love (and a punishment for her vanity);
- finally, as he sinks into greater oblivion despite Chandramukhi’s attempts to care for him after she abandons her profession, Devdas takes a last, aimless train ride across India.
Finally, as he had promised (“If it’s the last thing I do, I’ll come to you”), Devdas drags himself to the entrance of Parvati’s home – to which she has been restricted — where he dies just before she is alerted to the presence of a stranger’s body just beyond the massive gates that shut her inside as she runs to him. (While these details may spoil the story for a first-time viewer, it’s clear that most Indian viewers come to any telling of the tale with the plot well-known and its now-familiar highlights eagerly anticipated with each retelling.)
What I gather from the various discussions I've read is that Devdas is a story that romanticizes, perhaps not uncritically, its protagonist wallowing in sorrow and tragedy. So, Devdas Syndrome is roughly this: The romantic portrayal, of wallowing in sorrow, usually over an ended romance.
This is how I'm seeing the phrase used and the story discussed, primarily casually, across the internet. For example, this essay describes Devdas Syndrome as the desire to romanticize failure:
As a nation, we have always loved losers. And you cannot get a bigger loser than Devdas. He symbolizes everything that is wrong with us. (...)
Devdas refuses to rise above his sick and sodden self-image. In fact, he wallows in it and his
death is a supreme act of completely meaningless sacrifice. An escape,
as it were, from the sheer trauma of coping with life.
By enshrining the story of this eternal loser and packaging it so
lovingly and misleadingly as the story of the eternal lover, Bhansali
ends up romanticizing Devdas.
Or "The Devdas Syndrome: Turning Heartbreak into Hatred", which uses the phrase to describe people who cannot get over a failed romance, who are bitter and entitled and blame all their problems on the girl who told them no.
And of course, not every interpretation sees romanticizing tragedy as something to be avoid it. Some find it powerful and moving:
He wanted love and intensely loved, and yet could not express both.
His unfulfilled love did not stop him from loving uncondition- ally... thus making his love immortal even though it destroyed him later in his life. Ultimately, Devdas is a story that projects emotion as its hero.
This understanding of Devdas Syndrome seems borne out by this description of the author of the original 1917 novella feeling it has misled people
At one time Sarat told his pals that if he only he had visualised the negative impact his book was going to have on the youth he would not have permitted his book to be filmed at all!
(I do not know whether this anecdote is true, false, or a fictional portrayal, but in any event it's indicative of public opinion of the creation.)