What kind of tectonic plate activity exists on other planets in the solar system, or even large moons? Are there any deeper studies of what is needed for a planet or large moon to have tectonic activity, and how that activity might differ from planet/moon to planet/moon?
It's a huge field of research and a lot of interesting studies have been published over the years, but also a lot of speculation. Models and theories are mostly based on surface structures observed by remote sensing. Potential field data, as gravimetry can also add good constraints when available.
Tectonic forces might be rather common in large young planets, but as they cool the process usually stops. In our time and solar system, plate tectonics as we would recognize it, is only known on Earth (and maybe Titan). Other tectonic forces can cause similar features by local uplifts, volcanism and cryovolcanism.
Venus appears to have some kind of tectonic forces, at least in the past. An important part of the present research focus on the dynamics and chemistry of the Venusian mantle and how the topography helps us understand the development of the planet. The data from Magellan in the early 90's are particularly important. Studies of Venus also helps us to understand how tectonics developed on Earth. It's been suggested that lack of water as a lubricant in the Venusian upper mantle makes continental drift impossible, but mantle dynamics can still cause local/regional uplift and volcanism.
Europa is also supposed to have an odd kind stress that resembles tectonics, but caused by the tidal force from Jupiter that deforms the moon. We have the same tidal deformation on Earth, but it is much stronger on Europa and can cause dynamics that mix the interior. Somehow similar to how one should roll and press a lemon before cutting it to squeeze out the juice. Galileo is the main source of data, still.
Titan could be the candidate that most resemble Earth, but it's under debate and more data is needed.
The reasons to have plate tectonic on a planet are complicated and debated. I wouldn't say that we even have a good understanding of the continental drift on Earth yet. The best approach to understand what is needed for tectonism is to look at the planets energy budget, how much heat is generated and how can it leave the planets internal system. The chemistry of the mantel is also important, some dishes boil, other dishes fry. Water or other fluids appears to be crucial.
A rather interesting suggestion for Earth's unique plate tectonics is that it depends on additional energy harvested by the photosynthesis in green plants. It might be that life is the cause of mountains instead of the other way around.