What is wrong with storing deionized water in glass containers?
It is very difficult to make a container that does not introduce impurities to a liquid you place into it. As mentioned in the comments to the question, storing in a glass bottle will result in ions in your formerly pure water. 

This is thought to proceed by a mechanism called ion exchange (source):

$$\ce{ Glass.Na+ + H3O+ -&gt; Glass.H3O+ + Na+}$$

The Wikipedia article on dealkalisation states:


  Routine tests for surface dealkalization in the glass container industry all generally aim to evaluate the amount of alkali extracted from the glass when it is rinsed with or exposed to purified water. For example, dealkalization can be quickly checked by introducing a small volume of distilled water to a freshly made bottle and rolling the bottle gently to pass the water completely over its inside surface. The pH of the rinse water is then measured; untreated containers will tend to yield a slightly alkaline pH in the 8-9 range due to extracted alkali, while dealkalized containers tend to yield a pH that remains approximately neutral.


The article also mentions a technique called secondary ion mass spectrometry that can probe the composition of solid glass up to 1-2 nm depth below the surface. Leaching has been studied quantitatively in the context of storing nuclear waste in the glass state, see for example this PhD thesis (abstract in English, main text in French).

So storing in a glass bottle is not ideal because of contamination with ions. If you store it in a plastic bottle, on the other hand, you will find organic substance in your formerly pure water. Depending what kind of analysis you want to use your water for, different types of impurities will be more relevant (i.e. give spurious signals in the analysis). If you are analyzing samples with a GC/MS, glass bottles would be better (well, you would use an organic solvent, not water). If you are analyzing samples with atomic emission or absorption spectroscopy, plastic bottles would be better.

A first pointer would be the materials used in sample handling by the instrument itself.