If you just leave the paid tier, Rite.ly Pro and continue with Rite.ly Free, and if your DNS settings stay the same, your Link Ads will show until you hit the limit of link clicks for the free tier. Then the URLs will redirect to the original URL but not show the Link Ad anymore.
If you stop using Rite.ly and were using it with custom domain
If you completely go to another service and change the DNS for the domain, the Rite.ly links you have shared will not work. There is no way we can make the links work at all because the requests will no longer be coming to our servers.