If I use a regular GFCI in a Weather Proof In-Use box does the GFCI have to be the Weather Resistant kind?
Yes. If the receptacle is in a wet or damp location, the receptacle must be a weather-resistant type.
In a damp location, you can use any weatherproof cover. In a wet location, you must use an "extra-duty" cover. "Extra-duty" covers, provide weatherproofing even when a cord is attached to the receptacle.
National Electrical Code 2014
Chapter 4 Equipment for General Use.
Article 406 Receptacles, cord connectors, and attachment Plugs.
406.9 Receptacles in Damp or Wet Locations.
(A) Damp Locations ...All 15- and 20- ampere, 125- and 250-volt nonlocking receptacles shall be a listed weather-resistant type.
(B) Wet Locations.
(1) 15-and 20-Ampere Receptacles in a Wet Location. ...All 15- and 20 ampere, 125- and 250-volt nonlocking-type receptacles shall be listed weather-resistant type.
Chapter 1 General
Article 100 Definitions
Location, Damp. Locations protected from weather and not subject to saturation with water or other liquids but subject to moderate degrees of moisture.
Location, Dry. A location not normally subject to dampness or wetness. A location classified as dry may be temporarily subject to dampness or wetness, as in the case of a building under construction.
Location, Wet. Installations underground or in concrete slabs or masonry in direct contact with the earth; in locations subject to saturation with water or other liquids, such as vehicle washing areas; and in unprotected locations exposed to weather.
Install the GFCI indoors. Use its LOAD feature.
Since we're talking GFCI, the far better plan is to exploit the fact that GFCI devices are able to protect downline outlets.
Install the GFCI protection device indoors, somewhere appropriate such as the previous outlet on the chain, or as the breaker itself, or near the panel. You can use a "Deadfront" GFCI (A receptacle with no socket holes) if this circuit isn't allowed to have outlets in the GFCI location (e.g. bathroom circuits).
To make this legal you need a "GFCI Protected" sticker on all plain outlets thus protected.
@ user770395, Tester101 is correct but you also have to use a Weather Proof Cover when the receptacle can come into direct contact with water. I would use a In-use cover(1) but you could use a Weather Proof Cover but not a In-use cover(2). The in-use cover can still be shut and protect the outlet, while the not in-use cover can not.