Would that just be a reverse proxy? Is it even possible to hide a servers ip address with a public proxy? How would hiding the servers ip address work?
Services generally need to be in a fixed location as otherwise clients can't find them via DNS to connect to them. The servers that implement that service can of course be behind a proxy, which does indeed hide them as you say, and this is very common in load balancing configs. Whether that constitutes hiding their IP really depends what you're trying to achieve, since the balancer/front end is publicly visible, and very likely close to the back-end, in network terms.
If you want to hide the location of the service, rather than any of the collection of servers behind it, then you need to look at running a tor onion service, which resolves the location of the service in a way that doesn't reveal its IP, without using DNS in the usual way.