Learn Icelandic: Ég segi upp starfinu. - I give notice on the job.

In Icelandic, adding the word upp to segja creates the phrasal verb að segja upp. While it literally translates to "say up," it is the standard way to express that you are resigning from a job, giving notice, or canceling a contract or subscription.

Why is the word for job in the dative case (starfinu) here?

The verb phrase að segja upp forces the object that follows it to be in the dative case. Because of this rule, you must use the dative form of starf when you are quitting it.

What does the -inu ending on starfinu tell us?

It is the attached definite article, which gives us the meaning "the." The base word starf (job) is a neuter noun. When a neuter noun is in the dative case and definite, it takes the ending -inu to mean "the job" instead of just "a job."

Could I use the verb hætta (to stop or quit) instead of segja upp?

You can, but there is an important difference in meaning. Að hætta simply means to stop doing something. Að segja upp specifically means to formally resign or give notice. If you officially hand in a resignation letter, you use segja upp.