As a population health specialist, I see two issues that need to be addressed: first is the sheer volume of plastic waste generated with the use of nonrecyclable plastics—coffee and tea pods generally fall into this category. As this waste piles up in landfills, it causes environmental (and eventually human) harm. The second is whether the type of plastic contains any potentially harmful chemicals that can leach into something you’re going to ingest.
The reality is that plastic is impossible to completely avoid. However, I tell people to limit its use whenever possible. Storing food in plastic containers is fine, but heating that plastic creates an opportunity for chemicals to migrate into the food or beverage you plan to put into your body. Studies show that as they accumulate over time, these chemicals may interfere with normal functions of the brain and other organs.
There is not a lot of research on plastic coffee pods yet, but I would imagine the risk is probably fairly low because the hot liquid passes through quickly. That means there is not much time for leaching, which is how most children and adults wind up with chemicals from plastic in their system. Research associates these chemicals with a variety of health concerns, especially in children and newborns (from the heating of plastic bottles).
These coffee pods are convenient and probably safe, but shortcuts always come with shortcomings. If possible, find pods that are made out of compostable materials. Some companies are even making refillable metal or paper coffee pods.