Will cyanoacrylates damage plastic?
It's not likely, but first let's talk about "plastic".

There's two main kinds used on toys are PVC (polyvinyl chloride) and ABS (Acrylonitrile butadiene styrene). PVC is a bit flexible, and ABS is the rigid stuff (used to make harder shells, smaller children's toys, and stuff like LEGO blocks). You can generally tell something is PVC if it has a little bit of a give, or you can dig your fingernail into it slightly. 

Either way, most people I know consider super glue a safe fix for action figures and other toys. Aside from repairing them, some will apply a small amount to ball/socket joints to stiffen them up, or on top of custom paint jobs to act as a sealant.

Permabond, an adhesive company, even suggests that cyanoacrylates are ideal for PVC and also for ABS. However, ABS tends to me smoother and breaks more cleanly, so it may not be effectively bonded by super glue.

One thing to note about cyanoacrylates is that they have pretty low shear strength, which means if you're applying them to a part that's got movement, especially one that can be twisted, then the repaired piece is likely to just twist off.

An alternative is to use two-part epoxies, such as Devcon. They'll bond PVC and ABS both. However, they take a bit longer to dry and cure and they also have terrible fumes while they're being mixed, so you need proper ventilation. This bonds become much more durable to all types of pressure and use. They have not yet, in my experience, damaged a plastic. 

Another alternative for ABS plastic is to use a plastic weld solvent, such as bondene or plastruct brand. These solvents actually bond the separate parts together using a chemical reaction that fuses a thin layer of each part together. So, it's making the two broken pieces one fused piece, instead of two pieces with an adhesive between them. This is the same type of substance that toy manufacturers use when they are sealing toy parts together (such as the front and back of a figure's chest, or either half of a toy gun). This bond is fairly strong, and generally needs high pressure and some heat to break.