Greek accents have an undeserved reputation for being of no practical use. If sheer curiosity is not enough to persuade students to learn them, I hope these reasons might be more effective.
A knowledge of Greek accents:
1) helps you analyse previously unfamiliar vocabulary.
You are reading through the book of Acts and reach 1:18 where you have the phrase πρηνὴς γενόμενος. Many people are likely to be unfamiliar with the former word and so the question naturally arises as to whether it is the genitive singular of a first declension feminine noun *πρηνη. It is the accent which instantly tells you that this is impossible. Nor is this an isolated example. People who know the accents probably use them on a regular basis to decode and process unfamiliar texts.
2) helps you spot typos in Greek more readily.
Just as vultures hover over a battlefield, misplaced accents hover round other typographical errors in Greek. If you are an editor you probably will not check every ancient quotation, but not infrequently an accentual error is the first thing you notice to be wrong with a quotation and this leads you to check further. For example, if you know accents and you are reading Bart Ehrman's Orthodox Corruption of Scripture, p. 128, and come across ἐν ταῖς παλαίας ἀντιγραφαῖς, it may be that you instantly notice there is a missing iota in παλαιαις, or it may be that the wrong accentuation highlights the problem first. What if you're not sure whether there's a typo? After all, παλαιας is a possible word. Do you have to refer to an edition? No, the impossible accentuation on παλαιας confirms that there is at least one typo on this word. It's not therefore a big leap to infer that there are two. You're able to be pretty confident that the Greek letters are wrong without consulting an edition.
3) can be one of the fastest rough and ready guides to assessing the quality of someone's scholarship.
If a work contains lots of typos, it's pretty likely that references will be wrong too. If the author hasn't bothered to check such superficial errors as typos then s/he probably hasn't checked other things carefully either (e.g. quotations). The same applies to Greek accents. Carelessness in one area is likely to imply carelessness in others. The great advantage of Greek accents is that books generally contain more mistakes in Greek accents than in other parts of the text. So if you're wanting to check levels of scholarly care quickly you can scan a few pages with Greek in for 30 seconds and get a pretty good idea whether the Greek is carefully written. You probably wouldn't be able to get such a reliable impression of the typography of the English of a book in such a short time. Of course I need to issue the caveat that a scholar whose Greek was badly written could make a significant contribution. However, if you're going round the SBL book stalls and don't have long to decide whether a book by an unknown author is worth purchasing, a knowledge of Greek accents might provide an economic advantage.
So in certain circumstances knowledge of Greek accents could save time or even money. Are there any other significant practical reasons for learning them?