There is a slightly different take on the pour over coffee (filter cone). It is a device called the clever coffee dipper.
The real difference here is that you mix the water and the grounds in the filter cone, let it steep for 4 minutes, then drain the coffee through the filter. Normally I would think this is identical to pour over, however, while reading about french presses and Shabbat, I saw one opinion the mentioned mixing the grounds and then pushing the plunger is borer because the mixture is formed, then it is separated. I saw another article relating to tea, and filtering tea that makes me think it is OK (a tea pot with a filter on the spout).
What is the halacha here, and why? Can I use the clever?
Thank you for your interesting question. I'll start with the conclusion – in my opinion it is forbidden to use the "clever coffee dipper" on Shabbat. Now I'll try to explain the logic.
It is forbidden to filter mixtures on Shabbat that contain undesirable "waste", such as filtering wine from the dregs etc. So it would be forbidden, for example, to pour a mixture of tea mixed with tea leaves through a sieve or strainer in order to get clear tea from the mixture. Because of this many rabbis forbid using a teapot with a sieve in the spout if one is pouring out the tea that is mixed with the tea leaves. (If the leaves have all settled on the bottom of the teapot they are not considered "mixed" with the liquid tea on top, and there is no problem at all of pouring out the clear upper tea through the spout – the problem is when the leaves are still mixed and floating in the liquid tea). It is true that the Chazon Ish did permit using such a teapot, and much has been written to try and explain his seemingly problematic ruling. The general explanations are that when pouring a cup of tea from a teapot it is considered as "the way of eating", and not the forbidden labor of separating, (see Orchot Shabbat, 3,(79)). This logic (which itself is not accepted by all opinions – see Shmirat Shabbat 3, 62 [Hebrew, new edition]) would certainly not apply in the case of the "clever coffee dipper" – where the filter unit is placed on the cup in a specially designed way purely in order to filter the coffee.
So why do some rabbis allow using the coffee filters that are placed on top of the cup and water is poured through them? This is based on the ruling of the Shulchan Aruch (Orach Haim, 319,9 and the Mishna Brurah 33 ibid) which allows pouring water into a sieve that has grape dregs in it, in order that the water flow though the dregs (absorbing their taste) and out through the sieve. The logic here is that the water is poured into the sieve was not mixed, and it comes out not mixed. Based on this it is permitted to pour water over tea leaves (making sure not to break the laws of cooking on Shabbat) that are in a strainer, as the tea just flows through the strainer, absorbing the tea taste, and out the other side (see Shmirat Shabbat 3, 64 [Hebrew, new edition], and the Chazon Ish, Orach Haim, 53). This is exactly the use of the coffee filter that sits on the top of a cup. The filter cup has coffee in it, and a paper filter on its base. The water is poured into this filter cup and then drips through into the cup below after having absorbed the taste of the coffee beans.
However, the "clever coffee dipper" uses as different method, as you point out. The bottom of the filter cup is kept closed at the start of the process, so that the water can soak in the coffee beans for 4 or 5 minuets. Then, when the filter cup is placed on top of the drinking cup, it clicks open the bottom of the filter cup allowing the coffee to drip through the filter paper in the bottom of the upper filter cup. What is happening here is that first a mixture of hot water and coffee is made in the closed filter cup, and only afterwards in the bottom of the filter opened in order to start the filtering process. This is certainly forbidden on Shabbat, as one is separating the mixture by placing the filter cup over the drinking cup and clicking it open – it makes no difference whether the mixture was made 4 minuets ago, or 4 days ago, it is forbidden to use a filter sieve in order to separate the coffee from the dregs. Based on this understanding I see no way to allow the use of the "clever coffee dipper" on Shabbat.
I hope this is some help to you.