Can I break a voluntary fast for an Eid invitation?

First: no fasting on Eid day

Fasting is forbidden on the day of Eid al-Fitr and the day of Eid al-Adha (Sahih al-Bukhari 1990). The three days after Eid al-Adha, the days of tashriq, are also "days of eating and drinking and remembering Allah" (Sahih Muslim 1141), so most scholars say not to fast them either. See Can I fast on the day of Eid?

So the real situation is usually this: Eid al-Fitr has passed, you've started your six days of Shawwal (Sahih Muslim 1164) or a Monday fast, and someone invites you to a late Eid lunch.

A voluntary fast is flexible

• The Prophet ﷺ did it. One day he asked Aisha whether there was any food at home. She said someone had given them hays (a date dish). He said, "Show it to me, I woke up fasting," and then he ate (Sahih Muslim 1154).
• He approved of it for others. Salman visited Abu ad-Darda, who was fasting voluntarily, and told him to eat with him. The Prophet ﷺ later said Salman was right (Sahih al-Bukhari 1968).
• Accepting invitations matters. He said that when you're invited, accept; if you're fasting, make dua for your host, and if not, eat (Sahih Muslim 1431).

So you can either sit with them and make dua without eating, or break the fast, especially if your host would be hurt. Both are fine.

Do I have to make it up?

Scholars differ here. The Shafi'i and Hanbali schools say making up a broken voluntary fast is recommended but not required. The Hanafi school says you should make it up, and the Maliki school says you must if you broke it without a valid reason (many count a host's insistence as one). If you want to be safe, fast another day later; it's an easy extra reward.

This does not apply to obligatory fasts

If you're making up a missed Ramadan day or fasting for an oath, you should not break it for an invitation. Politely explain, join the gathering, and eat after sunset. See What are the voluntary fasts in Islam?