Is it safe to trust $_SERVER['REMOTE_ADDR']?

Is it safe to trust $_SERVER['REMOTE_ADDR']? Can it be substituted by changing the header of request or something like that?

Is it safe to write something like that?

if ($_SERVER['REMOTE_ADDR'] == '222.222.222.222') { // my ip address
    $grant_all_admin_rights = true;
}

Yes, it's safe. It is the source IP of the TCP connection and can't be substituted by changing an HTTP header.

One case you may want to be worry of is if you are behind a reverse proxy in which case the REMOTE_ADDR will always be the IP of the proxy server and the user IP will be provided in an HTTP header (such as X-Forwarded-For). But for the normal use case reading REMOTE_ADDR is fine.

$_SERVER['REMOTE_ADDR'] is the IP address the TCP connection came in on. While it is technically possible to bidirectionally spoof IP addresses on the Internet (by announcing foul routes via BGP), such attacks are likely to be spotted and not available to the typical attacker - basically, your attacker must have control over an ISP or carrier. There are no feasible unidirectional spoofing attacks against TCP (yet). Bidirectional IP spoofing is trivial on a LAN though.

Also be aware that it may be not be an IPv4, but an IPv6 address. Your current check is fine in that regard, but if you would check that 1.2.3.4 only occurs anywhere within $_SERVER['REMOTE_ADDR'], an attacker could simply connect from 2001:1234:5678::1.2.3.4.

Summarily, for anything other than critical (banking/military/potential damage >50.000€) applications, you can use the remote IP address if you can exclude attackers in your local network.

As mentioned above, it's not absolutely safe. But it doesn't mean, that you shouldn't use it. Consider combine this with some other methods of authentication, for instance checking COOKIE values.