With node-dota2/node-steam, you need an updated list of steam login servers, because the server list that ships with node-steam is like 9 months out of date. Periodically should also rebuild this list.

steam-client.js

const steam = require('steam');
const steamClient = new steam.SteamClient();
const steamUser = new steam.SteamUser(steamClient);
const dota2 = require('dota2');
const dotaClient = new dota2.Dota2Client(steamClient, true);

if (fs.existsSync('steam-servers.json')) {
	steam.servers = JSON.parse(fs.readFileSync('steam-servers.json'));
}

steamClient.connect();

steam-servers.json

[{"host":"155.133.254.132","port":"27017"},{"host":"155.133.254.132","port":"27018"},{"host":"155.133.254.132","port":"27019"},{"host":"155.133.254.133","port":"27019"},{"host":"155.133.254.133","port":"27017"},{"host":"155.133.254.133","port":"27018"},{"host":"162.254.195.47","port":"27017"},{"host":"162.254.195.45","port":"27019"},{"host":"162.254.195.44","port":"27018"},{"host":"162.254.195.44","port":"27019"},{"host":"162.254.195.47","port":"27019"},{"host":"162.254.195.45","port":"27018"},{"host":"162.254.195.46","port":"27017"},{"host":"162.254.195.45","port":"27017"},{"host":"162.254.195.44","port":"27017"},{"host":"162.254.195.46","port":"27018"},{"host":"162.254.195.46","port":"27019"},{"host":"162.254.195.47","port":"27018"},{"host":"162.254.193.47","port":"27018"},{"host":"162.254.193.7","port":"27018"},{"host":"162.254.193.6","port":"27017"},{"host":"162.254.193.6","port":"27019"},{"host":"162.254.193.7","port":"27017"},{"host":"162.254.193.46","port":"27017"},{"host":"162.254.193.6","port":"27018"},{"host":"162.254.193.47","port":"27019"},{"host":"162.254.193.7","port":"27019"},{"host":"162.254.193.46","port":"27019"},{"host":"162.254.193.47","port":"27017"},{"host":"162.254.193.46","port":"27018"},{"host":"205.196.6.67","port":"27018"},{"host":"205.196.6.75","port":"27019"},{"host":"205.196.6.75","port":"27018"},{"host":"162.254.192.101","port":"27019"},{"host":"205.196.6.67","port":"27017"},{"host":"205.196.6.75","port":"27017"},{"host":"205.196.6.67","port":"27019"},{"host":"162.254.192.108","port":"27018"},{"host":"162.254.192.101","port":"27018"},{"host":"162.254.192.101","port":"27017"},{"host":"162.254.192.109","port":"27019"},{"host":"162.254.192.100","port":"27019"},{"host":"162.254.192.108","port":"27017"},{"host":"162.254.192.109","port":"27018"},{"host":"162.254.192.109","port":"27017"},{"host":"162.254.192.100","port":"27018"},{"host":"162.254.192.108","port":"27019"},{"host":"162.254.192.100","port":"27017"},{"host":"162.254.196.68","port":"27019"},{"host":"162.254.196.84","port":"27017"},{"host":"162.254.196.83","port":"27018"},{"host":"162.254.196.67","port":"27017"},{"host":"162.254.196.68","port":"27017"},{"host":"162.254.196.84","port":"27018"},{"host":"162.254.196.68","port":"27018"},{"host":"162.254.196.83","port":"27017"},{"host":"162.254.196.84","port":"27019"},{"host":"162.254.196.67","port":"27019"},{"host":"162.254.196.67","port":"27018"},{"host":"162.254.196.83","port":"27019"},{"host":"185.25.182.77","port":"27018"},{"host":"185.25.182.77","port":"27019"},{"host":"185.25.182.76","port":"27018"},{"host":"185.25.182.77","port":"27017"},{"host":"185.25.182.76","port":"27017"},{"host":"185.25.182.76","port":"27019"},{"host":"155.133.248.52","port":"27019"},{"host":"155.133.248.53","port":"27018"},{"host":"155.133.248.50","port":"27017"},{"host":"155.133.248.50","port":"27019"},{"host":"155.133.248.52","port":"27018"},{"host":"155.133.248.51","port":"27018"},{"host":"155.133.248.53","port":"27019"},{"host":"155.133.248.53","port":"27017"},{"host":"155.133.248.52","port":"27017"},{"host":"155.133.248.50","port":"27018"},{"host":"155.133.248.51","port":"27017"},{"host":"155.133.248.51","port":"27019"},{"host":"162.254.198.130","port":"27018"},{"host":"146.66.152.10","port":"27019"},{"host":"146.66.152.11","port":"27019"},{"host":"155.133.246.68","port":"27019"},{"host":"162.254.197.180","port":"27019"},{"host":"162.254.197.181","port":"27019"},{"host":"162.254.197.180","port":"27017"},{"host":"162.254.198.133","port":"27018"},{"host":"146.66.152.10","port":"27018"},{"host":"155.133.246.69","port":"27019"},{"host":"162.254.197.180","port":"27018"},{"host":"162.254.197.181","port":"27018"},{"host":"162.254.197.42","port":"27017"},{"host":"162.254.197.40","port":"27019"},{"host":"162.254.198.133","port":"27017"},{"host":"162.254.198.130","port":"27019"},{"host":"162.254.198.133","port":"27019"},{"host":"146.66.152.11","port":"27017"},{"host":"146.66.152.11","port":"27018"},{"host":"155.133.246.69","port":"27018"},{"host":"155.133.246.68","port":"27017"},{"host":"155.133.246.68","port":"27018"}]

update-servers.js

var request = require('request');
var fs = require('fs');
var process = require('process');

request(
	'https://api.steampowered.com/ISteamDirectory/GetCMList/v1/?cellid=0',
	function(err, response, body) {
		var data = JSON.parse(body);
		var servers = data.response.serverlist;
		fs.writeFile('steam-servers.json', JSON.stringify(servers.map(function(s) {
			var [host, port] = s.split(':');
			return {host, port};
		})), process.exit);
	}
);