Perhaps the easiest way to wrap one's head around this theoretical framework is to first think of it in terms of video games that primarily use only a single category. Good examples of purely Drama focused video game genres are Walking Simulators (Gone Home, Dear Esther, etc.), certain adventure games (Dreamfall, Telltale's The Walking Dead, etc.) as well as most games that rely heavily on text to convey information to the player (link). Video games that are entirely about the Gaming aspect include a lot of fighting games (Street Fighter, Mortal Kombat, etc.), online card games (Heartstone, Scrolls, etc.), and pretty much any game you'd find in an arcade. Simulation is probably the most obvious to identify in that almost any game with the word "sim" or "simulation" in the title applies. Now that we've outlined the extreme cases lets take a look at the various instances when incompatibilities can occur.
Ever spent a long amount of time grinding only to find yourself wondering what the point of it all is? This is an excellent example of Gaming interfering with Drama. Oftentimes it's simply referred to as "padding," and usually involves the player having to complete a bunch of arbitrary tasks in order to advance the plot in a meaningful way. Conversely, intrusive unengaging cutscenes that detract from the player's enjoyment tend to be instances of Drama getting in the way of Gaming. More generally speaking there's a lack of player agency in these cases because the entire experience is railroaded in an unavoidable direction.
For examples of Gaming overriding Simulation (aside from puzzle bosses) look no farther than the Civ series of turn-based strategy games. Because of the complexity and scope of the franchise, the Gaming aspect has to act as a substitute for a lot of the intricate societal underpinnings. This sometimes results in ridiculous simplifications of the way human societies actually function. On the flip side War Thunder serves as a classic example of how Simulation can screw up Gaming. If you ever heard the expression "life isn't fair" it's in full effect here. Fighter craft such as the Japanese "Zero" were vastly superior to anything else at the beginning of WW2, but were completely outmatched by the end of the war. In competitive online spaces this inevitably leads to balance issues in which the only solutions are to break out the buffs/nerfs or create some kind of metagame constructs (like a battle rating) either of which further distorts whatever the simulation is trying to emulate.
Boss battles that are supposed to be epic, fall flat because the gameplay doesn't jive with the player's desired emotional experience. One-shot victories and bullet sponges are opposite extremes of this sort of problem. On the other hand invincible NPCs are a good way to illustrate how Drama can hamstring Simulation. One of the most iconic cases of this kind of thing has to be the death of Aerith in Final Fantasy VII. It's a scripted sequence that defies the rules established by the setting and is done entirely for story reasons.
The Threefold Model has gone on to given birth to GNS Theory and The Big Model, but both of these are fundamentally flawed in that they advocate the best option being to only focus on one paradigm when making a game in order to circumvent incompatibilities. In truth it's impossible to completely remove two of the three and still have something that would be enjoyable to play. Concessions have to be made, and that's not necessarily a bad thing because, despite all the conflicts the three modes have, they do sometimes support each other. Terms like "ludonarrative" and "emergent gameplay" are just a few examples of how one can complement another. It's not easy to do, or even well understood, but the Nemesis System in Middle-earth: Shadow of Mordor is proof that it's possible to do so successfully outside indie titles like Dwarf Fortress.