Nothing breaks. Every plugin and theme you have already downloaded and installed keeps working, permanently. Nothing stops, nothing locks, nothing phones home to check on you.
That is the important part, so it goes first. Your client sites are fine.
What actually stops
Two things, both of them about new files rather than existing ones.
Future updates for anything the pass was covering. Your installed version stays exactly as it is and keeps doing its job. It just stops receiving new versions.
New downloads from 99plugs.com. The library closes until you renew.
What keeps going
Everything installed. Forever. The GPL is what makes that true, and it is not something we could take back even if we wanted to.
Your individual purchases, which are unaffected and run on their own clock. If you bought a lifetime item key, that item keeps updating forever whether or not you have a pass. If you bought an annual item key, it keeps updating until its own year is up.
That last one surprises most people. A pass and an individual purchase are separate things, and letting the pass lapse does not touch the items you bought outright.
If you are letting a pass lapse deliberately
Keep your individual keys entered in the 99Plugs Update Manager. It is easy to clear the whole list out when a pass ends, and clearing it costs you updates you are still entitled to. See How to enable auto updates.
What to do about the plugins that stop updating
You have three options and they are all reasonable.
Renew. Buy the individual plugins you actually depend on, at $4.99 for a year or $9.99 for lifetime updates each. Or leave them where they are, which is fine for a site that is finished and not changing, and less fine for anything security sensitive.
Renewing
Your pass picks up where it left off and the whole library opens again. If you are renewing anyway, upgrading to lifetime is worth a look, because the discount is prorated from your original purchase date.