Rigorous academic arguments on whether AIs can replace all human workers?

There are a finite number of tasks that humans need performed to survive and prosper reasonably well. (Informal proof: There are a finite number of humans; some humans do prosper reasonably well; therefore, performance of some finite number of tasks is sufficient to allow these humans to prosper.)

To assert that robots or programs cannot do all of these tasks implies that there is at least one such task that robots or programs cannot perform. (I am choosing not to use "AI" as a noun, to bracket questions such as general intelligence or consciousness.)

I would ask, then, which ones?

but I have a hard time thinking of examples of the arguments being made in a rigorous way in the academic literature. This is surprising because it's a question with real practical import for humanity's future.

Why does that seem surprising to you? You are trying to forecast future technological development, and there is no general rigorous way of doing it.

The best you can get are negative statements. Some are strict impossibility proofs (perpetual mobiles, faster than light travel, etc.), some are more informal implausibility arguments, like the ones advanced by Tom Murphy on the Do the Math blog.

Positive statements in the form "we will develop technology X", where X is a technology that doesn't already exist at least in some prototypal form, are wild speculations. Historically, such predictions, even when made by domain experts, turned out to be wrong more often than not, and conversely actual innovations were typically not predicted decades in advance.

Regarding human-level AI, some people (Penrose et al) tried to put forward impossibility arguments, but these arguments are generally considered uncompelling and probably incorrect. Given the present understanding, it seems that AI is not theoretically impossible, but this tells us nothing about its practical feasibility.

I will go the brutal route and argue that, assuming nature's laws are computable in the computer science sense, we can simulate any given human brain, seeing as it is part of nature. By simulating this particular brain and arguing that the brain is the seat of intelligence in a human we have an AI in our hands thus confirming that AI is possible.

In other words, I think simulating a human brain and the possibility of AI are synonymous. Not necessarily in a strong sense. I realize this is not an academic argument, but might help you.

As far as I can tell, most academic work on this calls the question the "Church-Turing Thesis," or, more specifically, the "Strong Church-Turing Thesis." (As the SEP points out here, this is actually a completely new thesis, distinct from the actual thesis advanced by Turing.) This is regarded as an empirical, open question in academia, though almost everybody agrees that it is true. (AFAICT, this applies both to the regular and the strong thesis.) Many papers have mentioned the strong version of the thesis, but most thought about it deals with specific critiques, since the best positive arguments are simply to show AIs doing stuff people do. (There's also a lot of stuff that's flat out irrelevant see eg:

Nachum Dershowitz, & Yuri Gurevich (2008). A Natural Axiomatization of Computability and Proof of Church’s Thesis The Bulletin of Symbolic Logic DOI: 10.2178/bsl/1231081370 http://research.microsoft.com/en-us/um/people/gurevich/Opera/188.pdf)

An additional specific critique: some (eg Searle and Block) have argued that machines cannot have a mind. Eg Searle has argued that no machine can be conscious, with his Chinese Room argument:

Searle, John (1980), "Minds, Brains and Programs", Behavioral and Brain Sciences 3 (3): 417–457, doi:10.1017/S0140525X00005756, retrieved May 13, 2009

Similarly, Dreyfus has argued (in practice) that most human thinking is unconscious, and that it will never be possible to program a computer to execute these sorts of unconscious thoughts. He wrote a lot of articles and books to convey this and other critiques, but in general he claimed that early AGI researchers were making unwarranted philosophical assumptions- see here.

There are 2 ways to interpret the motivation of your question.

• If you want to know whether robots will replace ALL human workers, that seems equivalent to asking whether we'll "have" strong AI
• If you want to know if robots will replace vast majority of the human workers, see my comment here, I don't think you'll need academic arguments, as empirical data is starting to come it http://www.nytimes.com/2012/08/19/business/new-wave-of-adept-robots-is-changing-global-industry.html?_r=4&emc=eta1&pagewanted=all

an argument against based on Godel's theorem and similar math, what Turing called "The Mathematical Objection" and which has more recently been championed by Roger Penrose

Chapter 11 of Quantum Computing since Democritus is a succinct rebuttal to Penrose & company on this point. It also contains a nice steel-manning of Penrose.