Learn Finnish: Minun ei tarvitse korottaa summaa. - I do not need to raise the sum.

This sentence uses a necessity structure. When expressing that someone must or needs to do an action (using verbs like täytyy or tarvitsee), the person who has the need is put in the genitive case.

Since 'I' is doing the action, why do we use the third-person negative verb ei instead of en?

In necessity structures, the verb always stays in the third-person singular, even in the negative. Grammatically, the sentence works like 'It is not necessary for me to...', so we use ei tarvitse instead of en tarvitse.

I've seen tarvita used with a regular subject before, like minä tarvitsen. Why is it different here?

It depends on what follows the verb. If you need a thing (a noun), it works like a normal verb: Minä tarvitsen rahaa (I need money). But when you need to do something (followed by an infinitive verb like korottaa), it becomes a necessity verb requiring the genitive: Minun tarvitsee.

Why is the object summaa in the partitive case rather than the accusative?

Because the sentence is negative (ei tarvitse). In Finnish, the object of a negative action is always in the partitive case. If this were a positive sentence ('I have to raise the sum'), the object would be in the nominative: Minun täytyy korottaa summa.