Is 「持っていっていい」 just 「持っていってもいい」 with the 「も」 omitted?
I know 「てもいい」 means something along the lines of "may I" etc, is it the same for 「ていい」 or is there a slight difference?
Try translating the も as "still".
持っていっていい？ lit. Is it okay when I take it away/off?
持っていってもいい？ lit. Is it still okay when I take it away/off?
So, the nuance is the latter expects more possibility it could make inconvenience, thus asks more carefully on whether they don't mind. The difference is, however, minimized in affirmative/interrogative sentences compared to negative sentences.
来なくていい。 lit. It is okay when you don't come. (You're not needed!)
来なくてもいい。 lit. It is still okay when you don't come. (It's not compulsory.)
While they both can be translated "don't have to", the former is practically discouraging the hearer, and you may want to use the latter for neutral "may or may not" sense to avoid unnecessary misunderstandings :)