What chemical reaction can cause this blue-ish deposit?
Let's speculate a bit ;)


Colestyramine is a twodimensional polymer network, build from styrene and and 1,4-divinylbenzene. 
The polymer has a lot of benzyltrimethylammonium groups. That means it can act as an anion exchanger.
It is used to strongly bind bile acids
Neither the polymer itself, nor the adduct with bile acid are supposed to degrade in the bowel. The loaded polymer is supposed to be excreted.


The dishwasher tablets, on the other hand, contain the sodium salt of a maleic acid/acrylic acid copolymer.

That means we have two polymers, one with a lot of $\ce{Ph-CH2N^+(CH3)3}$ groups and another one with lots of $\ce{-COO-}$ groups.

Sounds like a mass wedding of two chain gangs :D

Edit

It is conceivable that this deposits a thin film on the spoons and the glassware.

In order to explain the bluish colour, we have two options:


The polymer aggregate absorbs in the 580 nm range - we observe the complementary colour. Due to lack of conjugation, I don't see how this could happen here.
The colour results from some dispersion effect related to opalescence, as suggested in Michiel's comment.