For the sake of clarification, because titles can only be so long: In one of the fights in is show, both sides charged into each other, forming walls of shields pushing and shoving against each other. In Knightfall, there was great emphasis placed on this, as it allowed the two opposing commanders to trash talk each other for 20-30 seconds. After a short while, one commander from side A ordered his formation to "break" in one spot, the relieved pressure making several soldiers from side B stumble forward into the gap and stagger behind side A's infantry line, allowing them to be killed, before the gap was closed and the shoving match resumed. For better clarification of what I mean, it's the battle in the forest in the final episode of season 1 (sadly I couldn't find a video clip, but it's on Netflix).
Is this a tactic that was actually used in warfare? I believe I've heard from posters on here that a similar sort of shoving match is hypothesized to have been used in hoplite phalanx warfare, but is it plausible that such a tactic persisted into the medieval era?
The short answer is "we don't know". The long answer amounts to "probably not".
Descriptions of how formation fighting actually functioned is rarely found and, when it is, the descriptions are so vague that they're useless on their own and a good deal of interpretation is required. The concept of othismos is a great example. The use of the word in both metaphorical and literal terms has led to continuing debate over whether or not Greek warfare prominently featured two sides shoving at each other with shields, using the full weight of the formation, or if it was intended to refer to a psychological push or aid to the hoplites and shield shoving wasn't a prominent part of Greek warfare.
(I'm simplifying here and haven't read all the literature, please don't crucify me!)
Even so, we have more to work with for Ancient Greece than we do for the High Middle Ages. Whereas those writing in Ancient Greece had first hand experience of fighting infantry battles and considered is a worthy subject to discuss, almost all of those writing about battles in the HMA were monks or priests, and the rest were members of the nobility and so had no interest on writing what an infantry battle was like.
This isn't to say there aren't any hints on the subject. For example, the Konungs Skuggsjá, which offers the clearest and most comprehensive description of infantry fighting that I'm aware of, nonetheless leaves the most important parts out:
If you are fighting on foot in a land battle and are placed at the point of wedge-shaped column (2), it is very important to watch the closed shield line in the first onset, lest it become disarranged or broken. Take heed never to bind the front of your shield under that of another (3). You must also be specially careful, when in the battle line, never to throw your spear, unless you have two, for in battle array on land one spear is more effective than two swords.
Apart from the importance of spears in infantry warfare, we learn that the shield wall shouldn't become disordered in the charge and that the right side of your shield should never be trapped under the left side of your neighbour's shield. This doesn't actually describe how men fought on foot, but it does give some important pieces of information about what contemporaries thought was important.
Now, to speculatively answer your question, I don't believe that the tactic used by the Templars in S01E10 is a likely one. There's just too much chance that the much deeper enemy formation will simply push through the gap and the Templars won't be able to close the gap in the line. If the enemy is pushing, then the whole lot of them will stagger forwards, putting a large number of bodies into the gap and, being armoured to a degree, won't be easy to dispatch. The result would be that the Templar line breaks and the superior numbers of infantry can attack the flanks and rear of their formation.