I'd like to use knotty pine boards as stair tread covers instead of actual tread covers. The tread covers I've seen are 1" or 2" wide pieces fit together to make a board, I'm guessing this is done to reduce warping? If I use standard pine boards, will I run into a warping problem?
Sure. One technique I've used (with 2x10 lumber, in my case) is to cut partial-depth lengthwise to give the board more flex in case of warping forces. So for my 2x10 treads I cut 1" into the board on the 1/3 and 2/3 points across the board.
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If you have exposed ends, stop the cuts short accordingly so they don't show.
If you're overlaying 1x boards, just glue them down well using construction adhesive. It would be wise to acclimate the boards in the home for a week or so first.
Be aware that pine is soft. It wasn't a problem in my home because we're a shoes-off family for the most part. One issue was when an HVAC tech bounced his dolly down the steps right after I built them, leaving a pair of dents on each nosing. I was not happy. That was the only damage they sustained during the decade or so I saw them in service.
Also be aware that varnished pine is slippery. I ended up installing a carpet runner down the center to remedy that. It's fine if you're aware of it, but guests won't be.
Like Kris mentioned in a comment, actual edge-glued or clear pine treads are available (though they're not cheap). They'll be 1" or 1-1/8" thick with a bullnose.