Drinking plain tea or coffee with no sweeteners or cream or anything else added is fine and won’t break your fast. I however always try to urge people to break their caffeine addiction, and while fasting is the best and easiest time to break it.
A 36-48 hour fast every week is an excellent idea. But if you want to strengthen your immune system you need to do a 3 day fast a few times a year (or more if you wish) and continually maintain a healthy diet.
Don’t forget the electrolytes as I describe in the faq, though if you’re going to fast for only 36 hours they’re not super necessary, but only as long as you get them from food.