Do the Buttons on Invisalign Have an Effect on the Teeth?

The buttons are bonded to the teeth in the same way that brackets are bonded to teeth. Just like brackets, when the case is complete the buttons must be removed. If the dentist/orthodontist is aggressive in the removal, the surface CAN be damaged, but that is the exception and not the rule.
Buttons are still better than brackets, in that most Invisalign cases only need a few buttons, whereas brackets are on ALL of the teeth. If bonding lead to damage, this would mean only a few damaged teeth instead of all of them damaged. But the reality is there is usually no damage at all.

The attachments are polished off at the end of treatment with no damage to the enamel. They are not snapped off like the old brackets which could damage the enamel. 

Invisalign buttons are made of composite, and therefore often difficult to distinguish between tooth when removing. Whe the dentist removes the buttons, they should take care not to remove tooth. I do this by using a polishing cup that removes the flowable composite quite easily while it only tends to polish the natural tooth's enamel. This wouldn't work as well if they use other types of composite than flowable. To be safe, they could err on the side of caution and leave composite and this would likely blend perfectly into the tooth anyway so it really should be easy for the careful clinician to remove the buttons.