Ultimately, it's going to depend on the AHJ (but doesn't everything?).
I've usually seen 110.26 cited in this type of situation. For this same reason, you're not allowed to use the space in front of a panelboard for storage (although I think that's a Fire Code issue as well).
110.26 Spaces About Electrical Equipment. Access and working space shall be provided and maintained about all electrical equipment to permit ready and safe operation and maintenance of such equipment.
I underlined the word "ready" because it could be argued that a dryer, while technically movable, is heavy enough and difficult enough to move that it does not permit ready operation and maintenance of the panel if placed in front of the panel.
110.26(A)(3) Height of Working Space. The work space shall be clear and extend from the grade, floor, or platform to a height of 2.0 m (6-1/2 ft) or the height of the equipment, whichever is greater. Within the height requirements of this section, other equipment that is associated with the electrical installation and is located above or below the electrical equipment shall be permitted to extend not more than 150 mm (6 in.) beyond the front of the electrical equipment.
As can be seen in this section, the working space extends all the way to the floor, not just directly in front of the panel. Again, one could make the argument that the dryer doesn't violate this section because it is movable. However, all the Inspectors I've ever met (which, admittedly, isn't all that many) have agreed that intent is for the space in front of the panel to be clear at all times. It's one thing if you build an electrical closet and then the owner decides to use it for storage -- that's on the owner's head if something goes wrong. But if you build it in such a way that you are requiring the owner to place something (like a dryer) in the panel's working space, then it's on your head.