I have been using plastic shoe boxes with transparent fitted covers or with plastic wrap covers, plus salad green boxes and other such transparent boxes for 5 years now with:
- never a box lost to disintegration into the soil
- never a problem with plant's roots
- with care about ventilation and placement such that plants would not overheat
- care to inspect the contents frequently enough to verify that there would be enough moisture in the box
- place drain holes through the bottoms of some boxes that contained plants that needed watering as opposed to misting with rainwater or distilled water
- care that tap water salts from the dissolved solids in most of our tap waters did not concentrate in the potting medium as moisture evaporated from the boxes
- care that boxes that were not adequately vented for pressure changes with changes in ambient temperature did not suck in misting water that was too high in dissolved solids, etc.
I would further say that I had to learn some of what I mention above the hard way, after inspecting plants, the contents of the boxes too infrequently.
Another thing to consider with transparent boxes is the tendency of algae, including some species that you might really not want, and also cyano-bacteria to grow in such boxes if the bases are exposed to light. Therefore, consider shielding the bases in many situations with aluminum foil wrapped up to the soil or planting medium line. If sunlight is to shine on the box during warm weather, avoid dark foils and avoid black paper, unless your objective is to cook the plants.
I acknowledge that there are some plant genera and species that would not do so well with too much light getting into the pot sides...so shield those roots from light with something that reflects light, unless you have a particular reason to need to warm up the roots. There are plenty of plants that like that, too, but don't let them reach cooking temperatures, which can be as low as 65 degrees for some species that I've grown or well over 100 degrees F for others. Get to know your plant's requirements, first. Some will want lids on the boxes and some won't.
Another point well made in response to the original question was that one can often use recycled containers. I have found reasons to use 1/2 gallon plastic milk boxes on their sides, plastic clam shell produce boxes, both those with ventilation slots and those without same, plastic meat boxes that have fitted transparent lids and black bottoms, etc. I have used cupcake boxes, both new and recycled, plastic cake boxes, etc. I only buy the new ones for use as very large community pots or very large germination flats.