Learn Spanish (Spain): No afines el piano ahora.

Because this is a negative command (telling someone not to do something). For the informal tú, negative commands use the present subjunctive form. Since afinar is an -ar verb, its ending swaps to an "e", making it no afines.

How would I say this if I were telling someone to tune the piano?

You would say Afina el piano ahora. Positive informal commands use the standard third-person present tense form (like él/ella afina), whereas negative commands force the vowel swap to the subjunctive (no afines).

How would this sentence change if I were speaking formally to someone?

If you were addressing someone with the formal usted, it would be No afine el piano ahora. For formal commands, you use the subjunctive vowel swap for both positive (Afine) and negative (No afine) commands.

What is the infinitive of this verb and are there other ways to use it?

The infinitive is afinar. While its main meaning is to tune a musical instrument or voice, you will also hear it used to mean to refine or to perfect something, like ironing out the final details of a plan.