Carbon has no effect on cycling what so ever so you can definitely take it out during the cycling period.
Now the question is do you need it when cycling is complete. There is two schools of thought on this, and personally I can see both sides of the argument have merit. The argument against carbon is that it really does not do much good except to remove medicines tannis etc (as already mentioned) and that carbon really only lasts about 6 days tops so there is really no need to put it in. The argument for carbon is that it does help to keep the water clearer, removes odours, and removes impurities that the dechlorinator does not. The reality is it all depends on the type of carbon you are using. Bituminous coal-based carbon is the one you want to use IF you decide to use it, it will not effect pH and is the same carbon used in Carbon Based drinking water filters, here is the article on it http
/www.drtimsaquatics.com/Article_db_folder/page41/page41.html. If you use Bituminous coal-based carbon it will help to remove some of the things in tap water that dechlorinators do not help with (BUT THIS IS NOT AN EXCUSE NOT TO USE DECHLORINATORS...you still need to declorinate your water) and it will help to keep the water clearer. Bituminous coal-based carbon also lasts longer than 6 days regardless of what others will tell you, I have done a lot of research on this, because I was previously on the fence about using carbon. Bituminous coal-based carbon generally lasts about 1 month (4-6 weeks), which is why most companies that make carbon inserts for filters tell you to change your carbon every 4-6 weeks.
Now as I said, I was previously on the fence about using carbon, however, I have now decided not to use carbon for 2 reasons. impurities in your water supply, you can either run your water through a filter (such as Britta or Pur) which I do, or you can ask for a water quality report from your water company to determine if there is something in your water that you should be concerned about (which is not a bad thing to have regardless of carbon use). The second reason I choose not to use carbon is for the simple fact that carbon takes up room in your filter, this is room that could be better utilized with things such as sponges, filter foam, ceramic pieces etc. These are things that allow beneficial bacteria to cultivate more readily than carbon. So I decided to go that route as more surface area for bacteria to grow = more bacteria in your filter which = a healthier tank.
Now should you decide not to use carbon on a regular basis I would suggest getting yourself some filter media bags, or even knee high stockings will work (clean of course) and a jar of Bituminous coal-based carbon should you ever need it to remove medicines etc. Marineland Black Diamond premium activated carbon is one example of Bituminous coal-based carbon - approximately $20 for 1.87L which if not using all the time will last a while, just make sure you keep it in a cool dry place.
Hope that helps.