Quite a few NES games had battery back-up saving (though LoZ may have been one of the first). I remember having a copy of Elite on the NES which had it...

SMB definitely doesn't though. Nor, for that matter, any of the SMB series on the NES. If you want to save your progress you'll need a copy either of SMB Deluxe on the GBC (which also had a nice little map feature, similar to SMB3) or Super Mario All-Stars on the SNES, which I think allowed you to save as well.