Will the YAG laser skin tightening treatment destroy the Juvederm I had injected 4 months ago?

The YAG laser penetrates the superficial layer of the skin.  If Juvederm was placed in this superficial layer, it would dissolve the Juvederm.  Typically, Juvederm is injected below this layer and would be safe from the YAG laser.  

No, those treatments are not believed to interfere with each other significantly.  I perform Yag laser tightening of the skin on the same day as hyaluronic acid filler treatments.  Just to clarify, the Yag laser can penetrate very deep about 1cm or less.  and filler is usually placed in the dermis of the skin which is a few or more mm down in depth;  therefore, technically the Yag laser does get to the same depth as this filler and deeper and dermatologic surgeons have an understanding of this.  A search of the literature, shows limited study of the interaction between these two treatments but one study did show no interaction. 

Thankfully, the laser is unlikely to change or harm your filler. Actually, many providers perform those procedures on the same day without any change in the filler effects. If it has been 4 months since your Juvederm injection, it is normal for your folds to be returning. Patients are usually ready for another syringe around 6 months.