This information is from my brother who teaches about wastewater treatment at a university and planned, built, and runs a treatment plant for a city. So I trust his information. The link below pertains to RVers and such, but the information would be similar for those of us on home septics. More than you wanted to know, but a relief after reading.
The following link is the worst case scenario. I think most formaldehyde based products are off the market because nobody wants them. The problem here is the concentration.
As I'm sure you have heard "the solution to pollution is dilution." In the case of the small amount of formaldehyde that will result from washing fabric I doubt there is any concern. Formaldehyde is an organic molecule that is food for some types of bacteria. The bacteria will break it down as long as the concentration isn't high enough to be toxic.
You could air out the fabric on a clothes line before washing it. That should lower the formaldehyde concentration prior to washing. Just smelling the fabric will give you a good idea of how much formaldehyde is present. I never realized what the smell was, but now have made the connection.
Long story short, not to worry. The septic will be fine with the little bit of formaldehyde you add from washing a few yards of cloth.