Is it permissible to eat with the left hand if the right hand is occupied with necessary tasks during a meal?

Eating with the left hand is reprehensible (makruh) unless there is a legitimate Islamic excuse. If there is an excuse preventing eating with the right hand, such as illness or surgery, then there is no reprehensibility in eating with the left hand. However, if there is no excuse, then whoever eats or drinks with the left hand, knowing of the prohibition, has disobeyed Allah and His Messenger.

If the tasks you are accomplishing with your right hand can be delayed, or their delay does not result in any harm, then be sure to eat with your right hand. However, if delaying those tasks would result in considerable harm according to Islamic law, then it is permissible to eat with the left hand.

Necessity would be, for example, paralysis of the right hand, or its fracture, or its burning, and similar circumstances that make eating with the right hand impossible. As for what some people do, drinking with the left hand for fear of contaminating the vessel, this does not justify drinking with the left hand.