I've played it since I was little, but don't remember beating it until this past Christmas. I did it without warp zones. I would consider beating the game with warp zones a cheat, since those are, I believe, intended to be save points.
Anyway, the trick is to not speedrun levels and to actively get coins. If you're collecting all the coins, you're going to have 10+ lives going into World 7 and 8. Some of the hardest levels in 7/8 have a green mushroom right before the hardest part, so you're never losing a life if you die repeatedly at that point. 8-3: you can speedily run under and over hammers with ease. The hardest parts are probably figuring out the mazes in the castles, especially the last one. After a long amount of frustration in the 8-4 castle, and not wanting to use the internet, I ended up asking my older brother who still had the game memorized from when he was a child.
My brother beat SMB1 the day I was born, he was 6, and as a reward my parents bought him a SNES.