All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is not forbidden to eat ice cubes unless there is a reason which necessitates them to be forbidden, like its water being impure and the like. However, eating ice cubes may lead to harm and in general this is not beneficial; therefore, one should avoid doing so.
Allaah Knows best.