What is the origin of the Italian expression 'a tutta birra' meaning 'at full speed'? Is it really related to beer?
No, it probably isn't. I've found that "a tutta birra" can be derived from a wrong translation of the French expression à toute bride, which should be translated as "a tutta briglia". "Briglia" means bridle, so the original expression is used to indicate that someone goes/something is done at full speed, like a galloping horse.
User @Federico found out an interesting fact. Comparing the frequency of the expressions "a tutta birra" and "a tutta briglia" in Italian books, using Google Ngram Viewer, shows that our beloved "a tutta birra" starts to appear around the '30s of last century.
Could it be related to the increasing use of cars which took place in those years? Here, we can just make hypothesis, but the use of this expression has been kept increasing for years, definitively surpassing the original "a tutta briglia" in the 70’s (here you can play with Ngram).
This interpretation can also be found in the book Acqua in bocca! Tutto quello che vorresti dire in italiano... come lo direbbe un italiano! by Roberto Bortoluzzi (Edizioni Casa delle lingue, Barcellona, 2015). This book also mentions that another Italian idiom closely related to the French idiom à toute bride (from which probably "a tutta birra" derives) is "a rotta di collo".
Are there other interpretations?
User @randomatlabuser enlightened me about another possible interpretation of the expression "a tutta birra". In fact, it seems that in the past horses were given beer as an energy drink, especially before going through hard parts of the travel. Also, modern diets for race horses include brewer's yeast, which is supposed to enhance the performance during the race (here it's briefly explained).