Developer program expires in 2 months, what happens if I publish a free app now?

Once your Apple Developer membership expires, all apps published under that developer's name will be removed from the App Store, regardless of whether they're free or not.

If you publish an app today, assuming it's approved, it will go into the App Store. Soon after your membership expires it will be removed. This might not happen on the exact day but it shouldn't take more than a few days after expiration.

From the Apple Developer page:

Expired Memberships

If your Apple Developer Program membership expires, your apps will no longer be available for download and you will not be able to submit new apps or updates. You will lose access to pre-release software, Certificates, Identifiers & Profiles, and Technical Support Incidents. However, your apps will still function for users who have already installed or downloaded them, and you will still have access to iTunes Connect and free development resources.

If your Apple Developer Enterprise Program membership expires, your apps will no longer be available for download and will no longer function for those who have already installed or downloaded them. You will still have access to free development resources.

iOS Developer University Program memberships do not expire.