What makes white phosphorus toxic?
There seems to be some difference on the development of symptoms depending on the route of uptake, but eventually poisoning with white phosphorus leads to kidney and liver damage.

Oral uptake leads to gastro-intestinal problems like abdominal cramps, nausea and vomitting, if you take up low doses chronically, this may lead to phosphorus laden stool (also called "smoking stool syndrome").

Since white phosphorus will ignite when it comes in contact with air, skin contact often leads to severe chemical burns. White phosphorus is also highly soluble in lipids and is therefore suspected to cross through our skin rather fast, leading to absorption in the body and subsequent poisoning by liver and kidney damage.

See reference 1 for a short overview and reference 2 for extensive information.

Regarding the exact molecular mechanisms is surprisingly little known and published. It is clear that poisoning with white phosphorus causes Triglycerides to accumulate in the liver, causing the fatty liver syndrome. Depending on the source it is also speculated that carbohydrate and/or protein synthesis in the liver is also affected (see for example here or here). Mostly this is explained by the high reduction potential of the phosphorus, which disturbs the fine balanced metabolic pathways. Finally this seems to leads to liver failure (see also reference 3).

According to the toxicology profile (reference 2) the damaged protein synthesis in the liver leads to fatty liver very fast.


  Liver damage in animals exposed to white phosphorus progresses
  rapidly. Four hours after receiving a single oral dose of white
  phosphorus, minimal fatty changes in hepatocytes were observed; by 12
  hours fatty changes were extensive (Ghoshal et al. 1969). Exposure to
  white phosphorus has been shown to damage the rough endoplasmic
  reticulum and cause a disaggregation of polyribosomes (Ganote and Otis
  1969; Pam et al. 1972). This damage results in impairment of protein
  synthesis, in particular, a decrease in the synthesis of the
  apolipoprotein portion of very low density lipoproteins (VLDL), which
  are required for the transport of triglycerides. A significant
  decrease in protein synthesis has been detected as early as 3 hours
  after oral exposure (Barker et al. 1963). The smooth endoplasmic
  reticulum is also involved in the formation of the VLDLs, and damage
  to the smooth endoplasmic reticulum also impairs the formation of
  VLDLs. The net result of these ultrastructural changes is an
  accumulation of triglycerides in the liver (Ghoshal et al. 1969). This
  results in steatosis and fibrosis, which is one of the mechanisms
  involved in the hepatotoxicity of white phosphorus. The mechanism
  behind the damage to the endoplasmic reticulum is not known; also, it
  is not known whether white phosphorus itself or a metabolite of white
  phosphorus is the damaging agent. In addition to these damages, white
  phosphorus or a metabolite causes damage to the mitochondria and
  nuclei in the livers of animals orally exposed to white phosphorus
  (Ghoshal et al. 1969). The damage to the mitochondria may impair the
  cell’s ability to produce ATP, thus resulting in necrosis of the cell.


Another possible (and additional) toxic effect may be traced to the (possibly enzymatic) generation of phosphines from phosphorus and water which are highly toxic to the central nervous system (here my source is only german, you can find it here). According to reference 4 phosphines are interfering with the COS gasotransmitter signaling. They can cause cardiac problems by this mechanism. Phosphines smell garlic like, this is an observation often made, when victims of white phosphorus poisonings are authopsied.

References


White Phosphorus Exposure
TOXICOLOGICAL PROFILE FOR WHITE PHOSPHORUS
Biochemical changes associated with acute phosphorus poisoning (in
humans)
Sulfurous Gases As Biological Messengers and Toxins: Comparative Genetics of Their Metabolism in Model Organisms