One is imperative ("Tirez", like instructions you'd see on a door telling whether to tirez or pousser), meaning "pull!". The other is infinitive ("Tirer") meaning "to pull".
It can also be equivalent to a noun-like "-ing" form (nominative? I don't know linguistics), as in "Tirer la porte c'est mieux que le pousser, si tu veux vraiment l'ouvrir." meaning "Pulling the door is better than pushing it, if you really wanna open it."
"Tirez" is the imperative form and should be accepted. However in French, it is usual to use the infinitive ("Tirer") rather than the imperative in things like recipes or instruction booklets or on signs such as might be on a door, indicating how best to open it. I suppose it is considered more polite, or at least less aggressive.
Still, I feel like it should be 'Tirez la porte' in this instance since we are being told (by the person in line behind us) to 'pull the door'. It would not be 'tirer la port' because 'To pull the door' is only a sentence fragment and a rather goofy thing to say outside of a Shakespearean play.
First congratulations on your great streak! In the "Type what you hear" exercise, I heard "Tirez la porte". I typed "Tirez la porte". Alas, Duo wants me to hear the homophone of tirez, i.e., tirer. Bad, bad Duolingo!! [Edit: Yay! "Tirez la porte" is accepted. "Tirez" is what it says on a lot of doors in Paris the last time I was there].
If you're in an office building, museum, bathroom, bus, or some public place that has a door that needs to be pulled to be opened, and you keep trying to push the door open, someone will eventually tell you to "pull the door."
(Or you can usually save yourself the embarrassment and read the sign that says "Tirer" on the door before it comes to that..)
"Tirer" or "Tirez" is what you would expect to see on a door in a public building that requires a pulling action to be opened. In English, one would expect "Pull." In French, just as in English, that word is not used for "close" or "shut." Rather, one would expect to hear "fermer la porte" for "close the door" and "ouvrir la porte" for "open the door."
*Both of these verbs would be conjugated in speech, of course.