Is there a taste or texture difference between gravity breweries and those that use pumps? I visited the UK's oldest licensed brewery, the Three Tuns, in Shropshire, which has been licenced since 1642. One of its key features is that all transfers can be carried out using gravity:

all raw materials are brought to the top of the tower, and then each movement of materials or fluids is carried out using gravity - not pumps

My question is: does this provide a difference in flavour, texture or other qualities to beers from those breweries that use pumps? I can imagine it might, as pumps cause agitation that won't exist in a simple pipe letting liquid flow downhill.
(additional information: the Three Tuns is well worth a visit - all the beers on show that day were delicious)
Directly not. Agitation won't change the properties of the beer enough as such. BUT... when pumping, it is harder to avoid oxygen/air being mixed into the wort/beer. Oxygen is very bad for the beer in most stages of the brewing process. So if you have a pumping system, it needs to be a good one that doesn't wirl air into the wort/beer.

The only stage in the brewing process where you actually want as much oxygen/air as possible is when pitching the yeast. At every other stage you will get oxidation which is one of the top reasons for off flavours, stuck fermentations, etc.