Does it harm juice if I store it in sub-zero temperatures?
Freezing juice will definitely not harm the juice itself, but it might harm your container if you're not careful.  The juice can expand quite a bit during the freezing process, so make sure that your container isn't completely full before you freeze it.

If the juice is already in a container (i.e. you bought it) then keep in mind that the container might not be designed for freezing and could explode at freezing temperatures due to the expanding volume.  If you're not sure, you might be better off transferring them to some large freezer bags.

I should also mention that freezing will only slow down the enzymatic activity that degrades fruit (and fruit juice) over time, not stop it completely.  Most fruit juice has been boiled to inactivate the enzymes, but if yours hasn't, then you might notice (bad) changes in the colour and taste over time.  If you plan to consume it within a few weeks, or if the juice already has various preservatives, then don't worry about it, but if it's very fresh juice and this is for long-term storage, then you might want to briefly boil it before storing if you're not sure about its history.

Keep in mind that the above will itself alter the taste, so if you plan to do it, test it on a small amount first to be sure that you can live with the "blanched" taste.