As we already know that alcohol consumption is Haram in Islam.
Can Muslims use beer Shampoo?
We know from Quran that drinking alcohol and eating pig/pork is prohibited/haram/haraam.
AFAIK it is allowed to wear jackets made with pig leather but not while praying.
An other basic rule says that anything which isn't clearly prohibited is allowed. From this point of view I'd say as long as you don't drink -by mistake- any kind of alcohol then using it for any other purpose should be allowed. In the following if I quote alcohol than the ruling should be considered the same for beer as well.
So I will try to answer the redefined Question:
Is using alcohol (especially beer) allowed, if we don't explicitly drink it?
Shia' View: The shia'a view I could find on this purpose is that alcohol is considered as a clean/tahir thing when the usage is not drinking and is therefore allowed.
This is based on this two fatwas in Arabic:
Sistani even says that alcohol of less then 2% can be drunk (in context of medicine I guess).
Al-Haeri on page 88, Question 40: says that beer without alcohol is halal to drink if it didn't have contact to alcohol. In Question 39 (same page) he allows cleaning lesions with alcohol!
Some scholars have said that alcohol is najis (some even made difference between physical najasa and abstract/moral najasa) while others consider it as tahir/clean.
Among those who consider it tahir are sheikh ibn 'Othaymeen, Imam a-Shawkani, Imam as-San'aani, imam al-Muzni (the student of Imam a-Shafi'i) and Imam al-Laith ibn Sa'ad.
But a big majority seems to consider it as najis, among them the Imams of the major 4 madhabs and the zahiri/dhahiri school.
Note that by physical najasa they mean that we must clean ourselves if we came in touch with this material.
The evidences for those who said it is only morally najis. There is a rule saying: any najis is haram, but not everything which is haram is najis" so alcohol is haram and not najis until there's a strong evidence for that. They also used the fact that Muslims have spilled it on the roads and markets when it was prohibited (see for example here) without cleaning the Jars etc. afterwards, so if it was najis they wouldn't have spilled it on the roads as one could conclude from this hadith. And they also added some comments on the evidences of their "opponents" saying that in verse (5:90) the defilement is a practical one not as this
is work of Satan
but not a defilement as for example neither gambling, nor [sacrificing on] stone alters [to other than Allah ], nor divining arrows are really najis. So the rule should be one for all the quoted things therefore alcohol isn't physically najis.
A fatwa on the najasa of alcohol (Arabic). A fatwa on the ruling of foodstuffs and beauty and personal-care products which have alcohol inside (Arabic and Urdu available)
But in any case -and I think both Sunni and Shia' scholars do agree about this point- the verse 5:90 says:
so avoid it
so one should avoid those things if possible.
So the answer is yes you could but should avoid it, because even if beer is considered as najis by some scholars you still usually wash the shampoo away and therefore clean up this najasa. And some say that beer is good for the hair.
And Allah knows best.