Most of what is being asked about is not a closed practice
Start with the distinction that does most of the work. Drinking matcha, brewing tea in a gaiwan, boiling chai with milk and spices, or pouring mint tea from a height are ordinary things that ordinary people do daily in the places they come from, and they are sold to anybody who wants to buy them. They are not restricted, they are not sacred, and treating them as untouchable is its own kind of misdescription — it takes an everyday drink and makes it an exhibit. A smaller number of things genuinely are formalised practices with institutions behind them. Chanoyu is taught within schools with lineages, licences and a curriculum, and performing it is a different act from whisking a bowl of matcha at home. The distinction is not between Japanese and non-Japanese people; it is between a practice with a teaching structure and a drink.
Three things that actually help
GET THE NAMES RIGHT, and use them. A chasen is a chasen, not a bamboo frother. Usucha and koicha are different preparations, not strengths of the same one. Gongfu cha is a method from a specific region rather than a general word for Chinese tea. Using the right word is the cheapest form of respect available and it also makes you better at the thing. SAY WHERE IT COMES FROM. "I make matcha the way it is made in Japan" costs nothing and is accurate. Claiming a tradition as your own, teaching it as an authority, or selling instruction in it is where the actual problem starts, and it is a claim about yourself rather than an act of drinking. BUY FROM THE PLACE WHERE POSSIBLE. Buying Uji matcha from a Uji producer, or Chaozhou pots from Chaozhou, puts money where the knowledge is. It is also, straightforwardly, how you get the better product.
Where care is genuinely owed
Some tea practices sit inside religious or ceremonial contexts, and some belong to communities with a specific history of having their culture taken, renamed and sold back to them. Where a practice is tied to a religious observance, or where the community is one whose material has been commercially appropriated at scale, the sensible posture is to learn about it rather than to perform it, and to be led by practitioners on what is welcome. That is a narrower category than the anxious version of this question assumes, and it does not cover most of what people are actually worried about. Drinking tea in a way you learned from another culture, accurately and with the name right, is how tea has moved around the world for two thousand years — every tradition on this site is downstream of somebody adopting somebody else's practice.
How TeaHQ writes about this, and why
Every tradition record on this site says plainly that it is compiled from published sources by people who do not practise it. That disclosure is required by the content rules rather than added as courtesy, and it is there because the alternative — describing a living practice in a confident anonymous voice — reads as authority that nobody has granted. The same posture is available to a reader. Knowing that your account of something is second-hand, and saying so, is most of what respectful use looks like in practice.