This "lump of labor" mental model has led The Guardian call for a reduction in the number of hours worked per week in the UK down to thirty, in order to supposedly address their record number of unemployed youths.
The same mode has also been used to justify union "job creation", or the hiring of more people to do unnecessary or unproductive jobs. (I must admit, one activity unions are highly productive at is Search Engine Optimization - pro-union pages dominate the top ranks of Google)
While this fallacy emerges frequently in the popular press at the behest of union leaders, it has been debunked many times already. From Henry Hazlitt's Economics in One Lesson (1952):
It is not only in reducing scheduled working hours that union policy has worked against productivity... Most of these policies have been followed under the assumption that there is just a fixed amount of work to be done, a definite "job fund" which has to be spread over as many people and hours as possible so as not to use it up too soon. This assumption is utterly false. There is actually no limit to the amount of work to be done. What A produces constitutes the demand for what B produces. (Chapter 19)
In other words, the potential stock of work to be done is constantly rising as worker income rises, because those workers now demand products they previously could not afford. Human wants are infinite: someone who supposedly has everything still wants new services in order to save time and have more leisure. Human lifespans are finite so time is always a limited resource, from the individual's perspective.
There are legitimate reasons to worry about technology and stratified worker productivity levels leading to unemployment and social divisions. However, those concerns should be framed through the lens of marginal worker productivity, not the over-worked lump of labor fallacy.