When reading Koine Greek, do I need to pronounce the accents? And if I do, how do I pronounce them?

Do you want practicality, or do you want historical accuracy?

Historical accuracy first. I’ve check Philomena Probert’s Ancient Greek Accentuation, and Vox Graeca. We know that the switch to stress accent must have happened by Gregory of Nazianzus (4th century): his poetry uses stress and not pitch accent as a base. We suspect that the Alexandrians were having to notate pitch accent because it was starting to die out; but we don’t know that for a fact. So the Gospel writers may or may not have had pitch accent, rather than stress accent.

Practicality:

1. Koine teaching does not use pitch accent: it’s quite a hurdle for people whose languages don’t have pitch.
2. The teaching of pitch accent for Attic or Epic Greek has not exactly been covered with glory (search “Yodelling Martians”)
3. Pitch doesn’t have as much of a functional load in Koine as it does in Attic, so there’s less motivation to teach it for practical reasons.

So you could use pitch accent, but I wouldn’t bother. Do accent, but keep it to stress.

EDIT: Per Philip Newton’s request: that would mean pronouncing all the different pitch accents (circumflex, acute, grave) the same way, as a stress accent.