Some sites block pings, some ISPs block pings as well so you might not be able to ping anything. Blocking ping requests or responses doesn't cause anything else to break but it also doesn't accomplish much either as you can just do a different kind of probe instead of ping to check for connectivity (but it's more expensive for the servers).

Some sites block all ICMP messages which actually causes real problems as they're part of the basic IP protocols. Blocking ICMP errors can cause extremely slow download speeds for users due to breaking the PMTUD protocol.