The most important water quality issues are related to declining water levels and increasing temperatures in rivers, lakes, and ponds, which in turn negatively impact fish and other aquatic organisms. The main issue is that the amount of oxygen that can be dissolved in water decreases with increasing temperature:
High temperatures and less water also mean an increase in evaporation, which increases the concentrations of many solutes. Some of these solutes, such as ammonia and nitrite, can be toxic at certain levels. Algal blooms can also increase during droughts, further robbing the water of oxygen. If oxygen levels drop far enough, fish are killed; there have already been a number of drought-related fish kills in Illinois this summer. While some fish re-populate fairly quickly, others can take several years to re-establish themselves.
Another consideration in urban areas is that many streams and rivers receive wastewater effluent. During droughts, runoff and groundwater discharge to these streams decreases, but wastewater effluent does not. So the percentage of wastewater in streams increases, which can lead to increased concentrations of nutrients and organic compounds, such as pharmaceuticals and personal care products. Some researchers have observed highly variable pH values during droughts, which can affect the toxicity levels of some compounds.
In 2005, the last drought of significance in Illinois, I was part of a water quality study of the Illinois River Basin. When we sampled in August 2005, the Illinois River was at a near record low flow. Dissolved solids levels were elevated for many solutes, such as chloride and boron. We could actually trace wastewater from Chicago all the way down the Illinois River to where it discharged into the Mississippi River (more than 300 miles). However, nitrate levels were actually very low in the Illinois River and most tributaries away from Chicago during the drought. Our conclusion was that the elevated temperatures and decrease oxygen levels were contributing to increased denitrification in the river. Researchers found similar results during droughts in the Meuse River in Germany.
The effects of drought on groundwater quality are much smaller than for surface water. In fact, in most cases there likely won’t be any negative water quality effects. Another reason why groundwater is superior to surface water! It is possible that declining water tables during droughts (some of which is due to increased pumping for irrigation) could cause some issues, mainly by allowing oxygen to penetrate to deeper depths, potentially promoting oxidation of iron sulfide minerals and increasing dissolved iron and arsenic levels. But I have not seen any actual research on this.