Tzippy,

If your move is a local one then usually you can leave the clothes in the drawers and movers would just wrap the dresser with moving blankets and move it as is.

For long distance move you would be required to pack the clothes in boxes. But again, it really depends on your movers - some would allow you to keep the clothes inside as long as the delivery would be quick enough. I would recommend you to pack the clothes in boxes for a long distance move, or inquire with your mover prior to booking what their policy is.

One word of advice - when moving locally some unethical foremen might try to force you to pack your clothes in boxes just to sell them (foremen usually get commission for selling boxes). In order not to fall into that trap make sure you verify with the representative of the company that clothes can stay inside, and have it written somewhere on the documents they fax back to you.