Wood – Why is the polyurethane finish I applied a month ago still tacky

The poly simply needs to cure for a while. Even after it "dries" and is safe to work on, the solvents that keep the polymers in suspension aren't all gone; they'll continue to evaporate, and the clear coat will fully harden in time. Check the can for your particular product; it should have a full cure time. It could take up to a month if you laid on a really thick coat and it's humid in your area.

Contrary to intuition, you can wax furniture that has a poly coat, much like you can wax a car that has a clearcoat. Just pick up a can of Minwax, apply and buff, then let dry overnight. This additional protective coating will be temporary, but hard, water-repellent and slick.

It's possible it's just the cool temperature and it will dry in a few days. It won't hurt to let it sit several more days and see if the tacky areas harden up on their own.

However my suspicion would be a compatibility problem between the urethane finish and the "green" mineral spirits. Although they probably say then can be substituted for real mineral spirits, I've heard of mixed results when using them for thinning oil based finishes. For a future project you could select all water-based finishing products which would have lower VOCs and would not require thinning with any solvents. Or use standard mineral spirits for thinning and save the "green" ones for cleanup.

If the finish doesn't harden on its own I would try to remove most of the finish from the tacky areas as you described. I would do it in stages with some drying time inbetween because if you remove too much you will likely also remove some of the stain resulting in inconsistent shading. In that case you may need to strip and refinish a larger area (or live with the spots).