Path of Exile's development team recently went through that stress, and in a new letter to the community, players and onlookers alike can see how the game weathered the transition. Here's the quick summary: Very well.
At the time of the letter, the game's peak concurrency was at 56,700 players, more than had been expected but very close to what the servers could handle. The game struggled a bit with patch downloads due to the CDN system immediately after launch, and there have been some crashes with various realms, but by and large the team seems happy with how the game is going and with overall sales figures. If you want more of the hard details on what a 30-hour open beta launch day looks like, read the full letter. And then get back to clicking things for money and loot.
[Thanks to tandarighttanon for the tip!]