Are the following sentences both correct?
- Je donne de la bière à Jean;
- Je donne de bière à Jean.
No, de la (as well as du, des and de l') is the partitive article. It is required when talking about things which - as a general but not absolute rule - can be counted. English does not require this. 'I have money'. 'I am eating fruit'. And so on. In French, that would be j'ai de l'argent and je mange du fruit.
The rules are:
masculine singular - du
feminine singular - de la
masculine or singular beginning with vowel - de l'
plural - des
So in your example, it would be, je donne de la bière.
However, if you're sitting there with a bottle in your hand and hand it to someone, it would be je donne la bière, but only because you're talking about one particular beer.