Why is polyploidy lethal for some organisms while for others is not?
Great question, and one about which there has historically been a lot of speculation, and there is currently a lot of misinformation. I will first address the two answers given by other users, which are both incorrect but have been historically suggested by scientists. Then I will try to explain the current understanding (which is not simple or complete). My answer is derived directly from the literature, and in particular from Mable (2004), which in turn is part of the 2004 special issue of the Biological Journal of the Linnean Society tackling the subject.

The 'sex' answer...

In 1925 HJ Muller addressed this question in a famous paper, "Why polyploidy is rarer in animals than in plants" (Muller, 1925). Muller briefly described the phenomenon that polyploidy was frequently observed in plants, but rarely in animals. The explanation, he said, was simple (and is approximate to that described in Matthew Piziak's answer):


  animals usually have two sexes which are differentiated by means of a process involving the diploid mechanism of segregation and combination whereas plants-at least the higher plants-are usually hermaphroditic.


Muller then elaborated with three explanations of the mechanism:


He assumed that triploidy was usually the intermediate step in chromosome duplication. This would cause problems, because if most animals' sex was determined by the ratios of chromosomes (as in Drosophila), triploidy would lead to sterility. 
In the rare cases when a tetraploid was accidentally created, it would have to breed with diploids, and this would result in a (presumably sterile) triploid.
If, by chance, two tetraploids were to arise and mate, they would be at a disadvantage because, he said, they would be randomly allocated sex chromosomes and this would lead to a higher proportion of non-viable offspring, and thus the polyploid line would be outcompeted by the diploid.


Unfortunately, whilst the first two points are valid facts about polyploids, the third point is incorrect. A major flaw with Muller's explanation is that it only applies to animals with chromosomal ratio-based sex determination, which we have since discovered is actually relatively few animals. In 1925 there was comparatively little systematic study of life, so we really didn't know what proportion of plant or animal taxa showed polyploidy. Muller's answer doesn't explain why most animals, e.g. those with Y-dominant sex determination, exhibit relatively little polyploidy. Another line of evidence disproving Muller's answer is that, in fact, polyploidy is very common among dioecious plants (those with separate male and female plants; e.g. Westergaard, 1958), while Muller's theory predicts that prevalence in this group should be as low as in animals.

The 'complexity' answer...

Another answer with some historical clout is the one given by Daniel Standage in his answer, and has been given by various scientists over the years (e.g. Stebbins, 1950). This answer states that animals are more complex than plants, so complex that their molecular machinery is much more finely balanced and is disturbed by having multiple genome copies.

This answer has been soundly rejected (e.g. by Orr, 1990) on the basis of two key facts. Firstly, whilst polyploidy is unusual in animals, it does occur. Various animals with hermaphroditic or parthenogenetic modes of reproduction frequently show polyploidy. There are also examples of Mammalian polyploidy (e.g. Gallardo et al., 2004). In addition, polyploidy can be artificially induced in a wide range of animal species, with no deleterious effects (in fact it often causes something akin to hybrid vigour; Jackson, 1976).

It's also worth noting here that since the 1960s Susumo Ohno (e.g. Ohno et al. 1968; Ohno 1970; Ohno 1999) has been proposing that vertebrate evolution involved multiple whole-genome duplication events (in addition to smaller duplications). There is now significant evidence to support this idea, reviewed in Furlong &amp; Holland (2004). If true, it further highlights that animals being more complex (itself a large, and in my view false, assumption) does not preclude polyploidy.

The modern synthesis...

And so to the present day. As reviewed in Mable (2004), it is now thought that:


Polyploidy is an important evolutionary mechanism which was and is probably responsible for a great deal of biological diversity.
Polyploidy arises easily in both animals and plants, but reproductive strategies might prevent it from propagating in certain circumstances, rather than any reduction in fitness resulting from the genome duplication.
Polyploidy may be more prevalent in animals than previously expected, and the imbalance in data arises from the fact that cytogenetics (i.e. chromosome counting) of large populations of wild specimens is a very common practise in botany, and very uncommon in zoology.


In addition, there are now several new suspected factors involved in ploidy which are currently being investigated:


Polyploidy is more common in species from high latitudes (temperate climates) and high altitudes (Soltis &amp; Soltis, 1999). Polyploidy frequently occurs by the production of unreduced gametes (through meiotic non-disjunction), and it has been shown that unreduced gametes are produced with higher frequency in response to environmental fluctuations. This predicts that polyploidy should be more likely to occur in the first place in fluctuating environments (which are more common at higher latitudes and altitudes).
Triploid individuals, the most likely initial result of a genome duplication event, in animals and plants often die before reaching sexual maturity, or have low fertility. However, if triploid individuals do reproduce, there is a chance of even-ploid (fertile) individuals resulting. This probability is increased if the species produces large numbers of both male and female gametes, or has some mechanism of bypassing the triploid individual stage. This may largely explain why many species with 'alternative' sexual modes (apomictic, automictic, unisexual, or gynogenetic) show polyploidy, as they can keep replicating tetraploids, thus increasing the chance that eventually a sexual encounter with another tetraploid will create a new polyploid line. In this way, non-sexual species may be a crucial evolutionary intermediate in generating sexual polyploid species. Species with external fertilisation are more likely to establish polyploid lines - a greater proportion of gametes are involved in fertilisation events and therefore two tetraploid gametes are more likely to meet.
Finally, polyploidy is more likely to occur in species with assortative mixing. That is, when a tetraploid gamete is formed, if the genome duplication somehow affects the individual so as to make it more likely that it will be fertilised by another tetraploid, then it is more likely that a polyploid line will be established. Thus it may be partly down to evolutionary chance as to how easily a species' reproductive traits are affected. For example in plants, tetraploids often have larger flowers or other organs, and thus are preferentially attractive to pollinators. In frogs, genome duplication leads to changes in the vocal apparatus which can lead to immediate reproductive isolation of polyploids.


References


Furlong, R.F. &amp; Holland, P.W.H. (2004) Polyploidy in vertebrate ancestry: Ohno and beyond. Biological Journal of the Linnean Society. 82 (4), 425–430.
Gallardo, M.H., Kausel, G., Jiménez, A., Bacquet, C., González, C., Figueroa, J., Köhler, N. &amp; Ojeda, R. (2004) Whole-genome duplications in South American desert rodents (Octodontidae). Biological Journal of the Linnean Society. 82 (4), 443–451.
Jackson, R.C. (1976) Evolution and Systematic Significance of Polyploidy. Annual Review of Ecology and Systematics. 7209–234.
Mable, B.K. (2004) ‘Why polyploidy is rarer in animals than in plants’: myths and mechanisms. Biological Journal of the Linnean Society. 82 (4), 453–466.
Muller, H.J. (1925) Why Polyploidy is Rarer in Animals Than in Plants. The American Naturalist. 59 (663), 346–353.
Ohno, S. (1970) Evolution by gene duplication.
Ohno, S. (1999) Gene duplication and the uniqueness of vertebrate genomes circa 1970–1999. Seminars in Cell &amp; Developmental Biology. 10 (5), 517–522.
Ohno, S., Wolf, U. &amp; Atkin, N.B. (1968) EVOLUTION FROM FISH TO MAMMALS BY GENE DUPLICATION. Hereditas. 59 (1), 169–187.
Orr, H.A. (1990) ‘Why Polyploidy is Rarer in Animals Than in Plants’ Revisited. The American Naturalist. 136 (6), 759–770.
Soltis, D.E. &amp; Soltis, P.S. (1999) Polyploidy: recurrent formation and genome evolution. Trends in Ecology &amp; Evolution. 14 (9), 348–352.
Stebbins, C.L. (1950) Variation and evolution in plants.
Westergaard, M. (1958) The Mechanism of Sex Determination in Dioecious Flowering Plants. In: Advances in Genetics. Academic Press. pp. 217–281.


(I'll come back and add links to the references later)