Learn Russian: Она хочет отомстить коллеге и навредить проекту. - She wants to take revenge on the colleague and harm the project.

The verb отомстить (to take revenge) is a verb of negative interaction or hindrance. In Russian, it directly requires the Dative case for the person you are taking revenge on, without any prepositions. Since коллега ends in -а, it takes the Dative ending -е.

Does навредить also use the Dative case in this sentence?

Yes! Навредить (to harm or damage) is another verb of hindrance. Just like giving or helping, harming something puts it on the receiving end of the action, which requires the Dative case. That is why the masculine noun проект takes the Dative ending -у to become проекту.

Is the colleague in this sentence a man or a woman?

It could be either! Коллега is a common-gender noun. Because its dictionary form ends in -а, it declines exactly like a feminine noun (taking the -е ending in the Dative case here) regardless of whether the colleague is male or female.

Why are отомстить and навредить in the perfective aspect here?

They follow хочет (she wants), and she wants to complete these specific, one-time actions with a final result. If she wanted to engage in a continuous habit of taking revenge or generally causing harm, we would use the imperfective verbs мстить and вредить instead.