You should check if loggedOn event's eresult is OK, otherwise there was an error logging in and therefore you can't proceed (request web session cookies). Replace: 
client.on('loggedOn', function(details) {
    client.getPersonas([client.steamID], function(personas) {
        console.log("============ Logged in ===============================")
        console.log('== Name: ' + personas[client.steamID]["player_name"]);
        console.log('== ID64: ' + client.steamID);
        console.log("======================================================");
        console.log("");
    });
    console.log();
    client.setPersona(1); // Shows the status as online. 0 = offline (It will still work though!) 1 = online
    setInterval(function() { client.webLogOn(); }, 1000 * 60); // Refreshes session every 10 minutes
});
with 
client.on('loggedOn', function(details) {
	if(details.eresult == SteamUser.EResult.OK) {
		client.getPersonas([client.steamID], function(personas) {
			console.log("============ Logged in ===============================")
			console.log('== Name: ' + personas[client.steamID]["player_name"]);
			console.log('== ID64: ' + client.steamID);
			console.log("======================================================");
			console.log("");
		});
		console.log();
		client.setPersona(1); // Shows the status as online. 0 = offline (It will still work though!) 1 = online
		client.webLogOn();
	} else {
		console.log(details);
		//Do whatever u want to handle the error...
	}
});
Also, you should rate limit this: 
community.on("sessionExpired", function(err) {
    console.log("## Session Expired, relogging.");
    client.webLogOn();
});
since it can be triggered too often and make steam temporary block your IP. You can do it like this: 
var lastWebLoginAttempt = 0;
community.on("sessionExpired", function(err) {
    if(Date.now() - lastLoginAttempt >= 30000) {
        lastLoginAttempt = Date.now();
        console.log("## Session Expired, relogging.");
        client.webLogOn();
    } else {
        console.log("## Session Expired, waiting a while before attempting to relogin.");
    }
});