Can the Removal of Lots of Tartar on my Teeth Affect How Effectively my Invisalign Works?

This totally depends on when your "retainer" was made.  A retainer should fit to the positions your teeth were in when the impression was taken for fabrication of the retainer.  If it was fabricated on a model of your teeth when you had "lots of build-up" and then subsequently you had the tartar removed the retainer will be loose.  If you suddenly loose 50 pounds....your old shorts won't stay up!  Having the tartar removed should not change the shape of your teeth but if your retainer had been stretched out by the tartar build up then you may be ready for a new retainer.  No matter what....you will be miles ahead without all of that tartar on your teeth.  Good Luck!

I assume the dentist that made the invisalign did not take an impression of tartar-covered teeth. Therefor, the aligners should still work. The best thing to do is talk it over with your treating dentist and see what he thinks.