is a foreign word, as is indicated even by the expression in the Alvismál--öl heitir með mönnum, en með Ásum bjór, ale it is called by men, by gods beer.
Vulgar Latin biber is from Latin bibere "to drink", from the Proto-Indo-European *peh₃(i)- "to drink". (The Latin b is explained by assimilation: the reduplicated *pi-ph₃-o- was voiced to something like *pi-bo- then assimilated to *bi-bo-.)
The suffixed form *poh₃-ti- became Latin pōtiō, pōtiōnis "drink", then Old French puison, meaning "magic potion". potion is also from the same Latin word thru Old French.
Bosworth and Toller say:
Beer, made from malted barley, was the favourite drink of the Anglo-Saxons. In their drinking parties, they pledged each other in large cups, round at the bottom, which must be emptied before they could be laid down, hence perhaps the name of a tumbler. We are speaking of the earliest times, for beer is mentioned in Beowulf
Seamus Heaney's Beowulf 480:
Ful oft gebēotedon bēore druncne
ofer ealo-wǣge ōret-mecgas
þæt hīe in bēor-sele bīdan woldon
Grendles gūþe mid gryrum ecga.
"Time and again, when the goblets passed
and seasoned fighters got flushed with beer
they would pledge themselves to protect Heorot
and wait for Grendel with whetted swords."
Here's another, ickier, occurrence, from a book of medicine:
Cockayne, Leechdoms, wortcunning, and starcraft of early England
genim beor dræstan ⁊ sapan · æges þ[æt] hƿite ⁊ ealde grut lege on ƿið omena gesƿelle.
"Take beer dregs and soap and the white of an egg and old groats, lay on for erysipelatous swelling."
Props to Language is the People's for talking about some cool brand names for the drink which gods call beer.