It is possible to hide public IP address from most Internet servers in most situations. This method involves an Internet service called an anonymous proxy server.

Proxy server

It is a computer that offers a computer network service to allow clients to make indirect network connections to other network services. A client connects to the proxy, and then requests a connection, file, or other resource available on a different server. The proxy provides the resource either by connecting to the specified server or by serving it from a cache. In some cases, it may alter the client’s request or the server’s response for various purposes.

Anonymous proxy server

In using it, all data sent to the service being used (for example, HTTP server in a website) must pass through the server before being sent to the service, mostly in unencrypted form. It is therefore possible, and has been demonstrated for a malicious proxy to record everything sent to the proxy: including unencrypted logins and passwords.
By chaining proxies which do not reveal data about the original requester, it is possible to obfuscate activities from the eyes of the user’s destination. However, more traces will be left on the intermediate hops, which could be used or offered up to trace the user’s activities. If the policies and administrators of these other proxies are unknown, the user may fall victim to a false sense of security just because those details are out of sight and mind
The bottom line of this is to be wary when using such servers, and only use them of known integrity (e.g., the owner is known and trusted, has a clear privacy policy, etc.), and never use proxies of unknown integrity. If there is no choice but to use unknown machines, do not pass any private information (unless it is properly encrypted) through the proxy.
In what is more of an inconvenience than a risk, users may find themselves being blocked from certain Web sites, as numerous forums and Web sites block IP addresses from proxies known to have spammed or trolled the site.