Does wrapping a bottle of whisky in plastic affect the taste?

Wrapping the boxes will not generate any heat (you need an energy source for this); it will however insulate what's inside, which means that changes in surrounding temperature are transmitted slower. When it gets colder in the room, the bottles will cool off slower than if they weren't wrapped; likewise, when the room gets warmer, they will heat up slower.

This is actually quite a Good Thing, fast fluctuations in temperature can cause all sorts of problems; if the glass heats faster than the cork, it will (may) expand faster which may end up in leaks, for example. Having said this, the plastic probably won't add very much insulation, unless you wrap it in a lot of thick plastic (bubble wrap, for example).

One thing to look out for when it comes to wrapping in plastic, is that moisture can make it a nice home for mold, which may affect second-hand value. Make sure there's room for breathing, or wrap in something that breaths (cloth or paper, for example).

There's more tips on https://www.singlemaltlodge.com/blog/item/how-to-store-whisky

Plastic isn't going to cause heating, and won't impact the quality of your whisky. Plastic wrapping is likely to trap moisture, however, which could be a concern as it could damage the labeling on the bottles. If you want to protect the boxes with plastic wrap my advice would be to make sure you have airflow.