With Moz Local, we do not actively do anything to your listings if you stop paying us. There are reports of Yext actively removing data when you stop paying, but since I don't work for them, I'm not comfortable commenting on their practices. Mihm is a far experienced in the local listings space than I am.
Here is what we say in our FAQs for Moz Local:
What happens if I cancel my listings with Moz Local?
Moz Local will simply report to the sites in our network that the listing is no longer under management by one of our customers. In this event, Acxiom and Neustar Localeze will revert your listings to their status prior to your Moz Local subscription. In some cases, your other listings will lose enhanced content like website URL, secondary category information, logos, and other images.
Moz Local will not actively remove your listing from our network of sites. You will always have the ability to reclaim your listings manually on each site if you decide to cancel your Moz Local subscription.