In general in Magic: the Gathering, "can't beats can":
101.2. When a rule or effect allows or directs something to happen, and another effect states that it can’t happen, the “can’t” effect takes precedence.
So Lich's Mastery and Platinum Angel prevail against Door to Nothingness. To win the game, you first need to get rid of the opponent's permanent. Or, in the case of Lich's Mastery, have a card that states that you win the game, like Battle of Wits. Lich's Mastery has a ruling below the card in Gatherer:
While you can’t lose the game, your opponents can still win the game if an effect says so.
and that's the reason why Platinum Angel separately states "your opponents can't win the game".
As a supplement to Glorfindel's answer:
104.3a A player can concede the game at any time. A player who concedes leaves the game immediately. That player loses the game.
In other words, even if you control a "can't lose the game" effect like Lich's Mastery or Platinum Angel, you can still lose by conceding.
No game effect can cause you to lose the game or cause any opponent to win the game while Gideon’s emblem is in effect. It doesn’t matter whether you have 0 or less life, you’re forced to draw a card while your library is empty, you have ten or more poison counters, you’re at your Glorious End, your opponent casts a second Approach of the Second Sun, or so on. You keep playing. (2017-04-18)
Other circumstances can still cause you to lose the game. You will lose a game if you concede, if you’re penalized with a Game Loss or a Match Loss during a sanctioned tournament due to a DCI rules infraction, or if your Magic Online® game clock runs out of time. (2017-04-18)
So a Gideon of the Trials emblem functions like Platinum Angel and stops both winning and losing, both Door to Nothingness as well as Approach of the Second Sun.
However, Lich's Mastery doesn't stop your opponents from winning, it only prevents you from losing. So even if you have a Lich's Mastery, an opponent can still win with Approach of the Second Sun. This is simply because winning is different from losing.