I am curious whether there are some teas that are not vegan (do not contain animal product, not even insects, fungal product is fine). I am interested mainly in teas available in European countries, but would appreciate information about any tea that does not fit the criteria. The tea should be non-vegan in itself, not becoming such by the way it is drunk somewhere (eg. adding milk to tea in UK).
Yes, there are a lot of teas (ie mixed infusions sold in tea shops) that are non-vegan and you should check the label and if it's unclear, ask staff for assistance before you buy.
I live in London. There is a nationwide chain of tea (and coffee) stores in the UK, Whittards, that sells a wide range of infusions. I have noticed that several of these products contain dairy. I've seen a few other fancy tea shops spring up with the same types of product, and I found even more non-vegan infusions. It's usually the sweet types; anything with a name like "caramel" or mentioning "creamy" seemed to contain dairy.
Honey is also used quite commonly in tea infusions, though this is more likely to be advertised in the name, thus easier to spot.
Another issue is the use of silk for making upmarket teabags, since unlike paper it does not affect the flavour. Fortunately this is generally advertised as a selling point, but I don't think manufacturers are bound to mention it by any regulation, and you might have to ask about that as well.
Any non-specified "flavouring", "natural flavouring" ("aroma" in German-speaking countries) is always suspect unless the product is labelled as suitable for veg* - and such are not uncommon in flavoured/"fruit" teas.
Insect contamination is never ever possible to completely preclude in herb-based products. If you are concerned about intentionally added insect products, stay clear of anything coloured with E120/cochineal (not unlikely with tea-like drinks), or using E904/shellac or maybe also E901/beeswax (both would be much less likely). Theoretically, teabag strings could be treated with wax, or casein glues could be used in the bag construction.
Some teas might be tested on animals, as was the case for Lipton tea products.