I wouldn't put a "length of fabric" piece in a binding with the selvedge on. The selvedge is actually stronger and a little thicker than the rest. But it will definitely shrink at a different rate. If fabric is washed with the selvedge on, you can easily see the slight puckering between the selvedge and the rest of the fabric.
I often leave the selvedges on when I cut WOF strips. When they are sewn with the 45º angle and trimmed (I line up the fabrics with no plus-sign pieces sticking out), there is about one molecule of selvedge still on the tips of each strip. And that molecule is in the seam allowance and won't be seen nor noticed. I even leave the white part on if it's not too wide. When it is folded and stitch, nothing shows.