You use this expression when you already did something and you don’t want to do it again. Basically, you did it once (or more), that’s it, and you don’t want anymore.

It implies j’ai déjà donné de ma personne.

Example:

• When we organize parties at school (for our club) we have to clean after. And since it’s basically the same who always clean, so when the question who want to clean came up, we said Non, c’est bon, on a déjà donné.

It’s also used when someone ask you something not hard/bad, but because you already did it once or more, you know that in fact it is hard.

• Tu veux nous aider à organiser le repas annuel ?
• Non, c’est bon je l’ai déjà fait l’année dernière, j’ai déjà donné.

This figurative expression is often preceded by a rhetorical question and used with the ironic merci. In a working environment, you can say something like:

Collaborer avec une marque d’une telle renommée mondiale comme Hermès ? Non merci, on a déjà donné.

• Suffering through a similar experience in the past, we’ve already had our fair share of nominal partnership with a leading brand in the field.

This sentence used to be a typical answer to someone collecting money on behalf of some charity or similar organization (e.g. la Croix-Rouge) by someone who was already asked by someone else from the same organization, and already donate money to it. The idea is to tell you are not refusing to donate, just can’t do it multiple time in a short period of time.

— Bonjour, est-ce que je peux vous demander de l’argent pour l’opération « Pièces jaunes 2019 » ?
— Oh, désolé. Je n’en ai plus. J’ai déjà donné.

The very same expression might also be used even when no initial donation exists, just to avoid telling you do not want or cannot afford to help.

As the other replies already stated, it can also be used figuratively to state you had some experience that cost or bothered you and do not want to repeat. This is likely by far the most common usage nowadays.

— Les crédits à taux variable, on a déjà donné !

— Les repas de Noël avec tes cousins, j’ai déjà donné !

— Travailler le dimanche, j’ai déjà donné !