Is it permissible to save some of the money designated for food during an Umrah trip fully paid for by others, and use it to purchase personal necessities?

If your friend gave you a sum of money for Umrah as an outright gift, then there is no harm in you saving some of it and spending it as you wish. However, if his commitment was only to cover the costs of Umrah and expenses during it, and the money he gave you was solely for that purpose, then you must seek specific permission for any remaining amount. This is because he did not give it to you as an outright gift, but rather for a specific purpose, to be spent on the necessities of Umrah.