It mainly depends of your PS2 version. I believe DVD-RW were not supported at all before V10 or V11.
But still, the PS2 dvd player sucks so much that you'll want to reconsider it before playing and DVD RW, as it WILL destroy your lens very quickly.
The type of disc isn't what matters. The difference in use of the laser is that movie discs read constantly, and the laser never really shuts off as all audio and video is streamed the entire time you watch the video. Game discs only read data as they need to, so once everything for an area is loaded, and the MIDI music starts playing, it shuts off the laser and stops reading the disc until more data is requested. The laser is only made to run for so many hours before it is expected to start burning out. Games may only use one hour of that time during every two hours of play, while movie discs will use nearly one hour every hour of play. All DVD discs read the same way, so there's no physical difference that would destroy the disc or the drive. The only difference is how often they are read.