The only time when coffee or tea is restricted for Muslims is during Ramadan, a month of spiritual fasting. From dawn until dusk, no food or drink is allowed, even water is forbidden. It's pretty clear-cut, with no gray areas.
More on Ramadan, from the About Islam Guide.
Latter-Day Saints (Mormons)
The LDS restrictions involving coffee and tea are fairly well known, though I'm sure most people don't really know the details behind it. Coffee and tea are specifically prohibited, as written in the Word of Wisdom: "Hot drinks are not for the body or belly" (D&C 89:9) It has been interpreted by the Church that Joseph Smith was referring to coffee and tea with this statement, as they were the only hot drinks commonly available at the time (early 1833). Some LDS members feel this is based on caffeine content, and then also feel that all caffeine is to be avoided. Some feel that the rule should be taken at face value, and therefore only refers to coffee and tea. The LDS church has no official position about caffeinated products other than coffee and tea.
More on the LDS caffeine issue, from the About Latter-Day Saints Guide.
There are no specific rules against either coffee or tea in Judaism, except for the broader rules of kosher eating. Both tea and coffee are kosher on their own, but there may be many other considerations to remember when enjoying your cup. One is flavoured coffee. Whether you are using flavoured beans, or adding syrups, you cannot be sure the kosher status of the many ingredients used to make these products. Many syrup manufacturers do offer kosher certified products. Another kosher issue can arise from decaffeinated coffee. Ethyl acetate is one chemical often used in the decaffeination process. One component of ethyl acetate is ethanol, which comes from grain. This would make coffee processed this way not kosher during Passover. There are several other aspects of coffee and tea that fall under kosher rules.
Star-K Kosher Certification has a good article with more information.
The Seventh-Day Adventist's believe strongly in the importance of a healthy and wholesome diet, free from alcohol, narcotics and other stimulants. Caffeine was once officially prohibited, but no longer. It is still recommended that caffeine be avoided.
More on SDA dietary guidelines
Like the Seventh-Day Adventist's, the Rastafarians follow a pure and wholesome diet. This excludes coffee, alcohol, salt, tobacco, meat and other processed foods. The foods eaten by Rastafarians (grains, fruits, vegetables) are all "ital" foods.
More on Rastafarianism, from the About Alternative Religions Guide
I am not an adherent to any of these religions, and I have done my best to understand the nature of these dietary restrictions. I apologize if I have used terminology improperly or represented any beliefs in an incorrect manner.