Learn Russian: Возьми либо чай, либо кофе.

То ли ..., то ли ... expresses the speaker's own doubt or confusion (like 'I can't tell if it's tea or coffee'). In this sentence, the speaker is offering a clear, deliberate choice to someone else, which requires либо ..., либо ... (or или ..., или ...).

Could I use или ..., или ... here instead of либо ..., либо ...?

Yes, Возьми или чай, или кофе is perfectly fine and means the exact same thing. Либо is just slightly more formal and often emphasizes that the choice is strictly mutually exclusive: you can only pick one of the two.

Why use the perfective imperative возьми instead of the imperfective бери?

The perfective возьми focuses on the result of a single, specific action: making your choice and taking the drink. The imperfective бери would sound more like a general invitation or continuous urging ('go ahead and take' or 'help yourself').

What case are чай and кофе in, and why don't their endings change?

They are in the accusative case because they are the direct objects being taken. For inanimate masculine nouns like чай, the accusative form is identical to the dictionary (nominative) form. Кофе is an indeclinable loanword, so it never changes its ending regardless of its role in the sentence.