The Shulhan Aruch (319:9) rules that pouring wine containing sediment through a strainer is prohibited and may constitute a Torah prohibition of Borer. Even though he is selecting the Ochel (food=wine) and discarding the sentiment, it is still prohibited, since he is using a utensil.
However, Maran also rules that it is permitted to pour water through a strainer containing wine sediment in order to give the water the flavor of wine.
The question is why Maran permits this? At some point the water mixes with the sediment and then is strained out. Shouldn’t this also constitute Borer? The commentators explain that since the water began the process as clean water and ended the process as a clear fluid, the fact that in the middle it was a mixture is irrelevant. In the first case it was prohibited since the wine was already mixed with sediment, and therefore straining it separated the good from the bad.
The practical application of this Halacha is using a type of tea kettle which contains hot water on the bottom and has a straining compartment near the spout to store tea leaves. When poured, the hot water passes through the tea leaves and makes tea. According to this understanding, this should be permitted, since the water begins and ends as a clear fluid, only temporarily mixing with the tea. This, is the position of Hacham Yishak Zafrani in his Va’Yizra Yishak.
Hacham Ovadia is even more lenient and permits using a kettle in which the leaves and the water are mixed together. As long as the leaves sink to the bottom, there is no problem pouring the tea out through the strainer at the spout, since they are not considered a mixture. However, when the water runs low, it will be a problem to continue pouring, as the strainer will be separating the water from the leaves.
There is also no problem of Borer in removing a tea bag from a cup of hot water (in a Keli Shelishi). The bag and the water do not constitute a mixture. However, there is a major discussion in the Poskim whether there is a problem of Borer in dangling the tea bag above the cup while the water drips out of the bag. This issue will be treated at length in a later Daily Halacha.
It is permitted to use a tea kettle containing tea leaves with a strainer at the spout.