Would everyone accept to work for free?
If 'work for free' means 'without being paid money', then the answer is yes.
I did not find even one research paper that would suggest that money is the only motivation to make people work. Most research suggests that correlations between salary and job satisfaction and between salary and work engagement are almost negligible. Internal motivations (like curiosity, interest in a specific topic, self-improvement, self-challenge, and so on) are much better predictors not only of job satisfaction and engagement but also productivity.
It is also worth considering that for the most part of human history, people worked without being paid any money. Salaried jobs (with salaries paid in money) became common only after the Industrial Revolution. And even with salaried jobs being the norm today, a significant amount of work1 is done without being paid: Domestic labour, child and elderly care, volunteer work, hobbies, and so on.
A lot of work is done not because it is rewarded monetarily, but because it needs to be done. For example, I clean my toilet not because I am paid to do it and definitely not because I enjoy it. I do it because it needs to be done to avoid future problems (and because I hate dirt more than I hate cleaning toilets).
Some other work is done because it relates to social or personal responsibilities. In modern societies, social responsibilities are often fulfilled by public agencies (e.g. maintenance of public property). In traditional societies, either some members do this work of their own volition or the community will achieve some sort of agreement on who and when does this work. Caring for children and the elderly is often a personal responsibility (although, some traditional societies consider it a communal responsibility) and all work done to fulfil this responsibility is done without pay.
And, of course, there will be a lot of work done because people enjoy it, receive social approval and respect, achieve self-fulfilment through it, and similar reasons.
There is a caveat. If we suddenly remove money from today's world and ask people to work without being paid, there will be a lot of people who will refuse to work, at least, initially. The number of those people will be most likely higher in cultures that assign high value to money and tend to see money as an indicator of personal success and fulfilment. I am talking here about cultural views and stereotypes, in other words, the perceived role of money in human lives. Whether this view is correct or not is a different topic.
If 'work for free' means 'without being rewarded in any way at all', then the answer is no.
What I mean by 'without being rewarded in any way at all' is that a worker does not receive any benefits, tangible or intangible.
There still be some people with a strong sense of responsibility who will work because they believe that it is something they ought to do. There will be some people who work because it is the lesser of the two evils (I hate dirt more than cleaning toilets). And, of course, all those people who work because they enjoy it, find it personally gratifying, fulfilling, etc. will continue working. However, there will be quite a few people who do not work at all because they are not being rewarded.
To be frank, I cannot imagine a situation where workers are not rewarded at all. Even housewives whose work is often undervalued, unrecognised, and looked down upon, receive at least some reward from friends and families complimenting their cooking, smart arrangements in the house, the aesthetics, and so on. Although, I do admit that this kind of reward can be insufficient motivation for many people.
Some additional notes
1. Culture and society
Even Marx, whose work Critique of the Gotha Programme made the slogan 'from each according to his ability, to each according to his needs' popular, was admitting that a direct transition from a capitalist to communist society is not possible. In the same work, he suggests that this transition will happen in two stages. At the lower stage, labour contribution to society will determine how much workers receive from this society. The transition to the higher stage (i.e. communism how it should be) becomes possible only when society and its members are completely free from the values and traditions of capitalism:
In a higher phase of communist society, after the enslaving subordination of the individual to the division of labor, and therewith also the antithesis between mental and physical labor, has vanished; after labor has become not only a means of life but life's prime want; after the productive forces have also increased with the all-around development of the individual, and all the springs of co-operative wealth flow more abundantly – only then can the narrow horizon of bourgeois right be crossed in its entirety and society inscribe on its banners: From each according to his ability, to each according to his needs! (Critique of the Gotha Programme. Part I)
Whether you agree with Marx or not, he makes a very good point: Moneyless economies require different cultural values and behaviours compared to money-based economies. I think that a moneyless society will put great emphasis on:
Collectivist cultures are more concerned with society's needs and goals. For example, collectivism was one of the best predictors of mask usage during the COVID-19 pandemic. Individualistic cultures tend to prioritise personal autonomy and may lead to the politicisation of questions, actions, and activities related to the public good (which could also be seen in the US where mask usage during the pandemic became a political issue and at the same time resulted in a higher number of super-spreaders compared to more collectivist cultures).
civic duties and social responsibilities
Civic duties refer to the responsibilities of citizens, e.g. voting and jury participation in democratic countries. Social responsibilities are responsibilities as a member of a group. There are a lot of things that people do not want to do for one reason or another. However, these things still need to be done. A person with a strong sense of duty and responsibility will be much more likely to do unlikeable but necessary things.
I also want to note that money is one of many reward mechanisms. Fame, recognition, respect, special treatment, and so on are examples of other reward mechanisms. In addition to rewards, societies employ coercion mechanisms to make their members perform necessary work. The social stigma attached to those who refuse to work can be used as coercion. So, while a moneyless society may not have one of the rewards mechanisms it will still have all other reward and coercion mechanisms. And even if people are not willing to accept working for free they might be forced to do so. The specific use of coercion and reward systems determines whether your moneyless society is a utopia or a dystopia. The absence of money does not automatically lead to any of these two outcomes.
It is a popular saying that capitalism stimulates innovation. I also read and heard many times that any non-capitalistic country will always and inevitably lag behind in innovation. I am not sure that this is entirely correct.
The USSR achieved remarkable success when it comes to space exploration or weapons despite being a socialist country with no free market. In all countries, capitalistic or not, most basic research is done using public funding (the state pays for expensive long-term research). Public funding is also essential for research in the fields that do not yield quick profit or will never be profitable.
Private business frequently focuses their research on consumer products that can bring profits and increase the company's value. They are not concerned with the public good and even if they are a responsible company and they have some considerations it is not their top priority or primary goal. It is even possible to argue that while capitalism indeed led to the increase in the quality of life short-term it is also responsible for the massive decrease in the quality of life in the future due to the harm caused to the environment.
A moneyless society can potentially be even more innovative than a capitalist society because theoretically people can freely choose their jobs and can unleash their full potential. At the same time, they can be more responsible when it comes to the public good and long-term effects of innovations because they are not concerned with profits.
I would be sceptical of anyone who makes any definite statements when it comes to innovation because we do not have any points of reference and/or data to support our conjectures. As a result, most predictions are based on ideology and assumptions about human nature2.
1 While some people equate work with paid labour only, this is not accurate. For example, according to OECD research, unpaid care work is one of the important factors explaining gender gaps in labour outcomes (participation, wages, and job quality). Across all regions of the world, women spend on average
between three and six hours per day on unpaid care activities, while men spend between half an hour and two hours. Even if we disregard the gender implications and focus on unpaid labour only, we will see that on average families spend between five and seven hours per day on unpaid care activities, which is only slightly shorter than a typical 8-hour workday.
Unpaid care work is defined in the cited document as 'Unpaid care work refers to all unpaid services provided within a household for its members, including care of persons, housework and voluntary community work (Elson, 2000). These activities are considered work, because theoretically one could pay a third person to perform them.'
2 If you visit discussion sites for US conservatives and US socialists/progressivists/communists this point will become very obvious. They use the same facts (or lack of facts) to come to completely opposite conclusions.