If a monk gets the tavern brawler feat, does his martial arts damage stack with the 1d4 damage given with tavern brawler?
For example: could a 1st level monk deal 2d4 damage with his unarmed strikes if he had the tavern brawler feat?
No, they don't.
From the Monk's Martial Arts feature:
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon.
From the Tavern Brawler feat (PHB, p. 170):
Your unarmed strike uses a d4 for damage.
Both abilities replace the unarmed strike damage die; they don't add to it. So if you have both abilities, you would still only roll 1d4 (plus Dex or Str mod).