Phoenix cards are pretty awesome in that they have the ability to come back after leaving the board, but at the cost of a lot of mana for pretty mediocre stats. Because of this, the creatures by themselves are slow to come out and pretty weak when they enter the battlefield. So everything here covers our main problems while letting the Phoenix play to its strength.
Draw/ discard power:
Thrill of Possibility is pretty self explanatory, discard a card draw two. Hopefully you get to discard a Flamewake or Sunstreak, pretty easy ones to bring back. Gate to the Afterlife and Celestus are pretty much the same in this category; draw a card, discard a card, gain a life, but Gate to the Afterlife has a much easier condition of triggering when one of your creatures dies. (This will be happening a lot with this deck.)
Searching:
Pyre of Heroes almost feels like it was made for a tribe like this. All of the Phoenix cards come back, so sacrificing them is no big deal, especially when you're bringing out more from the deck without paying their mana costs. You can sacrifice Flamewake every turn to fill up the board with Rekindling and Sunstreak, or if you need a bigger play sacrifice a 4 mana Phoenix to get Magma out and equip with the Basilisk Collar. Skarrgan Firebird is there at 1 specifically to sacrifice Magma if needed. Inventors' Fair can search out any artifact you need depending on the situation.
Mana:
Heraldic Banner is probably the most questionable card in the deck, but I enjoy it. It lets you tap for Red and give your Red creatures +1/+0 so it's helping out the deck in two needed ways. The Celestus does quite a bit, kind of the jack of all trades card in this deck. Nyx Lotus and Nykthos are definitely slow burn cards, but they can ramp for some good amount of mana if you get out a few Phoenix creatures with Pyre of Heroes. If you need faster mana ramping, I'd highly recommend dropping the Heraldic Banners for things like Mind Stone and Prismatic Lens before anything else.
Phoenix: I’ve narrowed this down to the Phoenix cards that manage to (mostly) pull their own weight. Older versions of this deck were built around Kuldotha Phoenix and his Metalcraft, so that's why the deck has so many artifacts. It works out well though since all the artifacts here really help bring out the best in these creatures. Flamewake Phoenix is probably the engine of this deck when combined with Pyre of Heroes. Easy to bring back from the graveyard with 1 mana and he gets out Rekindling and Sunstreak which have 4 power, which allows Flamewake to come back from the grave. Rekindling and Sunstreak fill the same roles here; strong enough for bring back Flamewake, they're the easiest/ most consistent 4CMC Phoenix cards to return from the graveyard, and they're perfect stepping stones to get to Kuldotha and Magma. Magma Phoenix is there to equip with Basilisk Collar and wipe the board, and as stated earlier, Skarrgan Firebird is there so you can sacrifice Magma with Pyre of Heroes. It can also be the biggest Phoenix in the deck if you get it out with Bloodthirst, but most of the time you won't be using this card. You can substitute it with Immortal Phoenix, but I prefer Skarrgan Firebird in case I have to discard cards.
Artifact Support: As mentioned earlier, the creatures we’re playing are far too weak on their own. Basilisk Collar fixes that problem easily, and since all of our Phoenix cards come back, we’re not too concerned on having them die in battle. Pyre of Heroes gets the creatures you want on the board for a lot less mana and is a great way to get Gate to get the draw power of Gate to the Afterlife on your own terms. Altar of Dementia is mostly here to kill Magma Phoenix equipped with Basilisk Collar at instant speed, but it can work wonders for just sacrificing something that would die anyway and mill your opponent (or yourself.) With all the Phoenix consistently coming back, this card can get an insane amount of usage.
Life Support: Basilisk Collar, Gate to the Afterlife, The Celestus, Inventors' Fair
Life gain is mostly a byproduct of the cards listed above. You certainly won't be going off the charts with the amount of life you get, but the consistency of the passive life increase you get with this deck will be noticeable and is quite nice.
Rising from the ashes: You should have a good idea of how this deck works now. All of the Phoenix cards either come back to the field or the hand, have flying, are annoying to deal with, and the artifacts take advantage of that. There's something oddly satisfying when your opponents get annoyed destroying the same creature over and over again.
I wanted to build a deck that fit the Phoenix theme of Immortality and overall I feel this is a solid build. It may require a few tweaks to fit your taste, but at the very least this can serve as a blueprint in case you were interested in building a modern Phoenix Tribal deck. Feel free to let me know your thoughts on the deck and what changes you would make. I realize this may not be fast enough to be competitive in the Modern format, but it’s a fun deck and definitely powerful if you’re playing with 3 or more players.