There are various synonyms for the phrase "open the door" that can be used depending on the context. For instance, "unlock the door" can be used when the door is locked and a key is required to gain entry. For a door that requires a push to open, "push open the door" would be an apt synonym. In situations where a door needs to be pulled, "pull the door open" would be appropriate. Similarly, "turn the knob" or "twist the handle" can be used for doors that have knobs or handles. In a more formal setting, "may I enter" can be used as well.