I haven't had problems for quite a while (browsers cleaned up their OpenGL implementations?) but the Planetarium could, conceivably, overtax the graphics power and possibly the computing power of your machine. This is more likely to happen if you set the Planetarium's zoom or time rate settings high or if you run another application at the same time that heavily uses 3-D graphics.
You should also be wary of web pages in other tabs that have a lot of advertisement activity. I find those can slow down my machine a lot, far more than the Planetarium can.
A problem does occur, here are some things you can do.