Learn Icelandic: Skál fyrir heiðursgestinum! - Cheers to the guest of honor!

There actually isn't one! While many formal wishes use a subjunctive verb (like veri or lifi), this sentence is a verbless exclamation. Skál is simply a noun (originally meaning "bowl") used as an interjection, much like "Cheers" in English. Not all wishes and exhortations require a verb.

Why is heiðursgestinum in the dative case? Doesn't fyrir usually take the accusative when it means "for"?

Fyrir takes the accusative when it means "for" in the sense of an intended recipient (e.g., buying a gift for someone). However, when making a toast in honor of someone, fyrir takes the dative. This is why a standard toast is Skál fyrir þér! (Cheers to you! - dative), not þig (accusative).

How is the word heiðursgestinum constructed?

It is a compound word made of heiður (honor) and gestur (guest). The -s- in the middle is a genitive ending that links the two nouns together (literally "guest of honor"). Finally, the -inum at the end is the masculine dative definite article, making it "the guest".

Where does the -i- in the ending of -gestinum come from?