Traditionally, this phenomenon is associated with an imbalance between the sex chromosomes and autosomes in polyploids diclinous animals. Definitely most of the higher animals is extremely limited in their ability to copy itself. On the contrary, such a property is widespread in higher plants and is necessary for reproduction polyploid mutants.

Two potential reasons for the greater degree of polyploidy in plants as compared to animals,

• Polyploidy can lead to infertility when breeding with non-polyploids. Plants however are somewhat able to overcome the disadvantage of this through self-fertilisation or vegetative propagation.
• Plant and animal early development is really quite different. It is relatively common for polyploid offspring to abort before birth in animals.

My favored hypothesis relates to the second one mentioned by James.  Developmental pathways of plants are highly compartmentalized and much simpler than they are in animals, so I think plant development is far more robust to genomic perturbations, like polyploidy, than is animal development.  I would predict, for example, that the rate of lethal mutation would be far less for plants than animals if you were to experimentally increase the mutation rate (e.g., with radiation).  Increasing the policy level most dramatically changes gene dosages, as opposed to amino acid sequences of proteins, and I think dosage changes are exactly the sort of thing that can disrupt animal development more severely than plant development.  Note that polyploid vertebrates generally represent more primitive lineages, like fish and amphibians.  I suspect developmental pathways became increasingly complicated in more recently derived vertebrate groups.

Great answers above!

I would add that in plants, more basal ("lower") plants such as ferns and some mosses are of higher ploidy level and have higher base chromosome number. More ancient lineages are predicted to have higher ploidy levels (more accumulation/history of whole genome duplications) based on their age if we look at polyploidy as a "seldom-reversible process" (Meyers and Levin, 2006). From one of my doctoral prelim answers, Spring 2014:

"Theory predicts that polyploidy should appear most frequently in organisms with low dispersal, large numbers of gametes and propagules, perennial habit, and self-compatibility or the capacity for asexuality/clonality, as these characteristics should increase availability of compatible mates early in establishment (Mable, 2004; Rieseberg and Willis, 2007). These predictions partly bear out in a comparison of polyploid speciation rates in angiosperms, ferns and lycophytes: Wood et al. (2009) found that 15% of angiosperm speciation events and ~31% of fern and lycophyte speciation events resulted in an increase in ploidy level. An alternative explanation (essentially a null model) for high mean base chromosome number (n = 57 for homosporous ferns compared to n = 16 for angiosperms; Soltis and Soltis, 2000) and rate of polyploid speciation events in ferns is their ancient origin: Meyers and Levin (2006) invoke a ‘polyploid ratchet’ model, as polyploidy is seldom reversible. This viewpoint gets support from the periodic increase in fossil guard cell size in basal angiosperms (Masterson, 1994)."

I've starred my favorite references on this question below.

*Choleva, L., and K. Janko. 2013. Rise and persistence of animal polyploidy: Evolutionary constraints and potential. Cytogenetic and Genomic Research 140: 151–170. (indeed, this whole issue! See below)

*Comai, L. 2005. The advantages and disadvantages of being polyploid. Nature Reviews Genetics 6: 836–846.

*Mable, B.K. 2004. ‘Why polyploidy is rarer in animals than in plants’: myths and mechanisms. Biological Journal of the Linnean Society 82: 453–466.

Masterson, J. 1994. Stomatal size in fossil plants: Evidence for polyploidy in majority of Angiosperms. Science 317: 910–914.

Meyers, L.A., and D.A. Levin. 2006. On the abundance of polyploids in flowering plants. Evolution 60: 1198–1206.

Soltis, P.S., and D.E. Soltis. 2000. The role of genetic and genomic attributes in the success of polyploids. In F.J. Ayala et al. [eds.], Variation and Evolution in Plants and Microorganisms, 310–329. National Academy Press, Washington, D.C., USA.

Stöck, M. and D.K. Lamatsch. 2013. Why comparing polyploidy research in animals and plants? Cytogenetic and Genomic Research 140: 75–78.

Van de Peer, Y., S. Maere, and A. Meyer. 2009. The evolutionary significance of ancient genome duplications. Nature Reviews Genetics 10: 725–732.

*Wolfe, K.H. 2001. Yesterday’s polyploids and the mystery of diploidization. Nature Reviews Genetics 2: 333–341.

Wood, T.E., N. Takebayashi, M.S. Barker, I. Mayrose, P.B. Greenspoon, and L.H. Rieseberg. 2009. The frequency of polyploidy speciation in vascular plants. Proceedings of the National Academy of Sciences 106: 13875–13879.

I think the most fundamental problem for polyploidy on animals is a problem of development.  As a very simple example,  manny horticultural varieties are polyploids of their wild ancestors.  These plants have no problem with in completing their life cycle with say,  multiple sets of flower petals.  This kind of development change tends to be much more lethal in animals -  back when I was a student,  animal breeders where experimenting with animals that produced double the set of muscles (through a kind of polyploidy breeding).  These animals had so many health problems that the program was abandoned. 

Well, good question. We all maybe can write a paper, I have seen a lot of responses, but it seems to me that answer you wanted to hear one related to how doubling genomes is possible in plants and not why? I like  thinking in the occurrence of genetic control in plant genomes that allows doubling genomes with no effects on gene expression. For example, imagine two gene communities that self-regulte or regulate each other expression so they can coexist in a cell. They probably synthesise all needs with no excess in anything. The imbalance of sexual chromosomes, and the signal they send, may be that signal im talking about, and that probably can cause disorders or development problems that limits the advantages of polyploidy observed in other organisms. But I think chromosome are more widespread that expected, just lets remember olympic athletes (women in particular), whose medals were retired after knowing that they have sexual chromosome disorders that made them superior in relation to their competitors. Anyway, I think that plants  have a mechanisms to buffer the genome duplication that animals lack.

Well, although the question was addressed in early 1960s and 1970s from genetics perspective by great geneticists of the past era, I wanted to find answers based on recent -omics advances as we get more intrigued in genome duplication events in plants and others. 

Thanks for great comments, Rafael, John, Keir (for a beautiful list of research articles!), Philip, James, Guy, and Yurij ! Great inputs, which may not be a paper, but the "ideas" would certainly end up somewhere in my blog !

By no means, this string is closed- and newer perspectives and insights would always be open !

I would add that polyploidy in animals is rampant in cancers and suggest that the underlying mechanism may be related to that observed in plants. Please see attached. Your criticisms will be greatly appreciated.

Thanks!

Article The shock of being united: Another lesson from plants?

In addition to all of the above, it is only in the Angiosperms among plants that recurrent polyploidy is common. A major factor in this is that unreduced gametes in angiosperms have a high mean frequency of about 0.6%, and this can rise to about 27.5% in hybrids.

Such unreduced gametes may lead to triploids or tetraploids (reviewed by Leitch and Leitch 2012), and may be repeated to form higher polyploids (Oliver 2012). Polyploidy (whole genome duplication) is common and is highly tolerated in Angiosperms, unlike in Gymnosperms. Mostly ancient whole genome duplications have also occurred occasionally in metazoans, including the vertebrates, but metazoans are generally far less tolerant of such abnormalities than are angiosperms.   

Leitch AR & Leitch 2012 Ecological and genetic factors linked to contrasting genome dynamics in seed plants. New Phytologist doi:10.1111/j.1469-1837.2012.04105.x

Oliver KR 2012 The Dynamic Eukaryote Genome: Evolution, Mobile DNA, and the TE-Thrust Hypothesis (Chapter4). PhD Thesis, Murdoch University