Learn Finnish: Barista paahtaa kahvia entistä enemmän. - The barista roasts coffee more than before.

Entistä is the partitive form of the adjective entinen (former, previous). In Finnish, you can express "than something" by putting that something in the partitive case before the comparative word. So, entistä enemmän literally means "more than the former", which translates naturally as "more than before".

Could I use the word kuin (than) to say "more than before" in this sentence?

Yes, but the phrasing changes. Instead of entistä enemmän, you would say enemmän kuin ennen. Using the partitive entistä directly before the comparative is a very idiomatic and compact way to express this without needing kuin.

Why is kahvia in the partitive case instead of the accusative?

Coffee (kahvi) is an uncountable mass noun. Because the barista is roasting an undefined amount of it, and the action is an ongoing process, the object must be in the partitive case.

What is the infinitive of the verb paahtaa, and does it have consonant gradation here?

The infinitive is also paahtaa (to roast), which is a type 1 verb. It keeps the strong consonant grade -ht- in the third-person singular here. You only see the weak grade -hd- in closed syllables, such as the first-person singular minä paahdan.