While I could not find any polls of economists verifying whether they considered the theory that the amount of labor is fixed to actually be a fallacy, I was able to find the Survey of Americans and Economists on the Economy which randomly surveyed 1,510 members of the American population and 250 people with PhDs in economics, back in 1996.
The survey does not ask specifically about the lump of labor fallacy, but it asks enough questions about other topics go give us an informed guess as to what their opinion about it might be.
When asked "Generally speaking, do you think each of the following is good or bad for the nation's economy, or don't you think it makes much difference?", few economists spoke negatively about: more women entering the workforce (2%), increased use of technology in the workplace (0%), trade agreements between the U.S. and other countries (3%), nor the recent downsizing of large corporations (16%). That's not consistent with a group of people who believes labor is zero-sum.
For example, the Luddites opposed the increased use of technology in the workplace because they feared it would leave them without a job. Yet, none of the polled economists (i.e., 0%) oppose increased use of technology in the workplace. In fact, when asked if technology displacing workers was a reason the economy is not doing better than it is, only 2% would call it a major reason, with 24% calling a minor reason, and 74% saying it was not a reason.
Similarly, while the majority of Americans (54%) believe that trade agreements cost jobs to United States, economists overwhelmingly disagree. When asked "Do you think that trade agreements between the United States and other countries have helped create more jobs in the U.S., or have they cost the U.S. jobs, or haven't they made much of a difference?", 50% of economists respond that trade agreements have helped create jobs and 42% respond that they have not made much of a difference. In fact, only 5% of polled economists believe trade agreements have cost the United States jobs.
Additionally, when asked if companies sending jobs overseas and downsizing were part of the reason the economy is not doing as well as it might otherwise be doing, the majority (57% and 58%) claimed it was not a reason. Only a small minority (5% and 6%), claimed it was a major reason.
Another case where the idea that labor is fixed can be a factor is immigration, as @matt_black mentions in his answer to Is there a link between rising UK youth unemployment and immigration from Eastern Europe? According to the survey, 47% of surveyed Americans think that "There are too many immigrants" is a major reason why the economy is not doing better than it is. To that 47%, you have to add another 32% who believe that it is a minor reason for why the economy is not doing better than it is. By comparison, only 1% of economist will list it as a major reason and 19% will list it as a minor reason, which puts the number of economist who think it is not a reason at 80%. Furthermore, when pressed on what was the primary factor that is holding back the economy, the percentage drops to 0% for economists (versus 7% for the general public).
Finally, when asked "Some people say that these are economically unsettled times because of new technology, competition from foreign countries, and downsizing. Looking ahead 20 years, do you think these changes will eventually be good or bad for the country or don't you think these changes will make much difference?", 93% of economists believe these changes will be good for the economy in the long term while only 2% think it will be bad for the economy.
In Summary
While I could find no poll asking specifically about the lump of labor fallacy, the result of the Survey of Americans and Economists on the Economy are not congruent with a group of people who believe there a fixed amount of labor. Generally speaking, economists favor free-trade, replacing workers by technology, and the participation of women and immigrants in the workforce. Moreover, they don't view either companies downsizing or sending jobs overseas if it's more profitable, as detrimental to the economy.
In fact, in the long term, practically all economists think these things will be beneficial to the economy.
Therefore, I will conclude it's extremely likely that the majority of economists believe the claim that the amount of work available to labourers is fixed to be fallacious.
