In Thailand everyone has very high respect for the king. Practically every business has his picture on the wall and he is celebrated many times of year. And it's not just made up stuff, the people really do love and respect him. That's why, for example, if you drop a coin you should never stop it with your foot because that would make you stomp on the kings picture and it will be considered really rude by everyone who sees it. With so much respect from all the people, I don't think there is anything the king can do for it to be really insulting. Not that I think he would ever try.
That being said, the king almost never takes part in any political issue. It's for the government to handle, not the king. That's why neither side, not even the red shirts, will insult the king. He doesn't have any part in it and despite the internal conflicts, everyone loves the king. If someone really is insulting him, it's not for political reasons and hence can't be censoring political speech either.