At Daylight Atheism, Ebonmuse gives a cogent and mostly accurate critique of Communism.
To be pedantic, Ebonmuse criticizes "Communism" in the sense of the practices and policies of self-described Communist governments, such as the Soviet Union, China, North Korea, Cambodia, Vietnam, Albania, etc. ("practical Communism"). This is a completely legitimate target of criticism, and a completely legitimate way to use the unqualified word "Communism", but there are other more theoretical constructions that do differ substantively from the practical sense.
Furthermore, there are two distinct senses of theoretical Communism: economic and political. Political Communism is, even in theory, usually totalitarian or crypto-totalitarian, but political Communism is separable from economic Communism. Care should be taken, I think, to understand that a legitimate and (mostly) accurate critique of practical Communism cannot be applied willy-nilly to its theoretical cousins, especially its economic-theoretical cousin.
Communism lacks a good mechanism to allocate resources to where they are most needed, resulting in waste, shortages and inefficiency.
To be more precise, it is the part of Communism that specifies a state-planned economy that results in inefficient distribution of commodities. The problem is practical (or information-theoretical); it's not per se a problem in Communism's fundamental economic assumptions. In theory, a planned economy should be more efficient than a distributed free-market economy, but in practice information-theoretical laws of diminishing returns and exponential complexity of large formal systems doom the notion of central commodity distribution planning.
The information-theoretical analysis does not, however, entail that a state-planned capital allocation system would necessarily be worse that the Capitalist alternative; capital allocation is centralized under Capitalism in the capital-owning elite. Furthermore, a mixed system, where capital was partly state-owned and partly worker-owned, might have the advantages of both central and distributed planning, and would not violate the spirit of economic Communism.
Communism discourages productive effort and innovation.
This criticism is accurate, but it applies only to certain phases of practical Communism, and the identification of the specific cause as the lack of (immediate) "material rewards for invention, innovation, or greater productivity" is controvertible.
The two largest Communist governments, the Soviet Union and China, experienced massive economic growth and industrialization in the early phases of communist rule. (Admittedly, a large percentage increase from a tiny base is not at all impressive to anyone with a basic understanding of statistics.) There clearly there was some operating motivation other than immediate material reward. Additionally, there are any number of examples in Capitalist economies (notably the open-source software movement) of productive endeavor without immediate reward.
Furthermore, most productivity is rewarded by nothing more than brute survival in a static or contracting Capitalist economy (and quite a lot even in an expanding economy). While simply removing the work-or-starve ethic seems insufficient (and its replacement by a work-or-get-shot ethic seems worse), its replacement with a less brutal ethic would seem desirable from a humanitarian standpoint.
I suspect it's far more likely to lay the blame for late-stage practical Communism's lack of innovation on the failings of specifically political Communism. We see precisely the same lack of innovation in non-Communist totalitarian societies as well as in many large-scale Capitalist corporations (with an internal political culture of monarchical feudalism).
Communism necessarily denies the freedom of the individual.
There's no doubt that Communism as practiced has massively and inexcusably denied individual liberty and freedom, but it is again the specifically totalitarian component of political Communism, which politically privileges a specific party, that entails the denial of individual freedom.
But economic Communism does not necessarily entail the denial of freedom. Indeed, under ideal circumstances, everyone is free of economic coercion; they are entirely free to apply their productivity to suit their personal preferences. Even under non-ideal circumstances, people would be expected to have at least greater choice in allocating their personal productivity as desired.
Since economic communism does not permit exploitation or economic coercion, it has to be expected that some people will choose to be unproductive or less productive than they would have been under Capitalist work-or-starve economies. Since we have never seen economic communism in practice, it is an open question whether a great many people would choose to be negatively productive, and whether those who do choose negative productivity are those who would be very productive if coerced.