I've tried a variety of things, including checking the door with every possible combination of lit mushrooms, but none worked. The only thing that happened here was a random encounter with (IIRC) Glaze, who killed me very quickly. I never managed to get a rematch, however.
You must complete the game and dodge every name in the special thanks section of the credits
When you reach the end of the game, there will be one last bit of the game where you are given infinite health with which to try and dodge all of the names in the Special Thanks portion of the credits. If you successfully manage to dodge every name in the credits in a single run, you will be able to pass through the door in your screenshot.