Can we digest raw starch?
We eat raw starch, in small to moderate amounts, almost every day without any obvious side effects. 

It's difficult to find good figures, but bananas contain at least 1% starch (if ripe) and, I estimate, as much as 5% assuming only completely yellow bananas are eaten raw.

Fresh garden peas contain around 2% starch given that 40% of the sugars are converted to starch (estimated using the sugar content of snow peas).

Eating raw starch (in moderate amounts) is good for you

As the Wikipedia starch article states,


  Digestive enzymes have problems digesting crystalline structures. Raw
  starch will digest poorly in the duodenum and small intestine, while
  bacterial degradation will take place mainly in the colon.


This is not necessarily a bad thing since undigested starch, or resistant starch as it is known, is a source of insoluble dietary fiber. Upwards of 80% of any raw starch consumed will remain undigested. This overview notes that there is 'also limited evidence for a prebiotic effect of some starches that resist digestion'.

Counter-indications

The same article referred to above states that '[m]ost resistant starches [...] are thought to be fermentable'. And foods fermenting in the gut will produce gas so you might want to avoid eating excessive amounts of raw starch to avoid uncomfortable digestion problems. 

This article from a raw food site warns against possible toxicity problems,


  Neither salivary amylase (ptyalin) nor pancreatic amylase can commence
  digestion of the starch until it is released from its globule. These
  starch-containing globules are, therefore, not digested at all and
  must be eliminated from the body as so much debris. Undigested
  materials such as these are toxic in the body and pose an eliminative
  burden without providing energy or other value.


I haven't yet found any scholarly literature to corroborate this, and the fact that this article fails to mention the, well established, role of resistent starch as dietary fiber makes me suspect its accuracy. 

TFD's answer above states,


  The energy loss from easting raw starch is huge, there is little
  point. Eat it cooked.


Although I haven't attempted to verify this, I can't see how energy loss counts against eating raw starch if you are getting enough calories from the rest of your diet. It doesn't seem to count against eating celery. The raw food article has another explanation for why we choose not to eat very starchy foods raw,


  Most starches just don't taste that good in their raw state.
  
  Carrots, sweet potatoes and yams are notable exceptions, however,
  because these tubers, in addition to containing starches, also contain
  enough sugars to give them a sweet flavor.


It seems clear from this article that at least some people enjoy potatoes raw. It also suggests that it is, for the most part, safe to eat them although the resistent starches, as I also indicate, might cause you some gastro-intestinal discomfort.

Another answer here seems to suggest that raw potatoes contain more toxins than cooked. This is incorrect unless the potatoes were cooked at temperatures over 170°C (340°F). The toxins in potatoes are low to begin with unless they are allowed to develop green patches, in which case cooking at normal temperatures won't help.

It is worth noting that, while eating raw starch is probably not going to give you any big problems, you should check that there are no other problems related to eating raw foods that you are not accustomed to.