It seems that this action is permissible, and there is nothing wrong with it, for the following reasons:
1 – Most of the people who come to these sites know that this name is not a real one, rather it is just a nickname for the writer.
2 – This is not counted as lying, because the people who run these sites do not stipulate that people have to use their real names.
3 – There is no reason why a person should not give himself a nickname or kunyah, and use this nickname or kunyah instead of his real name, and this is not regarded as lying.
4 – Using one’s real name may cause problems for some people, so they may chose to use a screen-name or a kunyah etc.
5 – Hiding one’s real name may serve a purpose, such as if the daa’iyah is well known and if his name was disclosed the people of bid’ah would refuse to speak to him because they know how strong his arguments are, or he may be banned by the owners of the site and so on.
Hence there is nothing wrong with entering sites and chatrooms with a screen-name.
And Allaah knows best.