I'm talking about the classic paper tea bags (sold empty) that you can buy and then put whatever you want in; sometimes different sizes are available for cups vs teapots, etc. Yes, I do realise most "pre-filled" tea bags like Lipton have bags made from roughly the same material, but I don't know if those don't filter out nutrients too.
Nutrients are microscopic, the pores in tea bags are far too large to act as a filter. However, tea bags will absorb a very small amount of liquid, and in that liquid there will nutrients, sugars, colors, volatile compounds and other things in suspension. So, the bag is going to absorb some of the tea, but the amount is negligible.
That doesn't mean that everything will get out of the bag, though. Even though the pores in the material are too big to trap nutrients the material is going to impede water circulation. Also, when the leaves expand they tend to get constricted by the bag, which further reduces flow. If you stir the bag around or squeeze it you'll often see darker water coming out of it, which demonstrates this effect. The upshot is that the material isn't filtering vitamins and minerals, but the slower circulation means some tea is going to be left inside the bag, and therefore nutrients along with it. Stirring the bag around a bit helps to a certain extent.
Paper filters such as tea bags, exclude at a size of 1 to 1000 microns, nutrient vitamins and minerals in foods and drink are measured in nano sizes for consumptions. Unless you are dealing with a unique or bulk formulation a paper filter should not limit the vitamin and mineral passage from one location to another.