Earth has about five million times as much mass, so the Sun cutout would have about a fifth of a millionth Earth's gravitational pull.
EDIT: In the grand spirit of what-if.xkcd.com I think it behooves us to take a humorously dark scientific look at this.
A larger problem would be 1.14×10¹⁸ kg of cardboard suddenly falling onto Earth's surface. Aside from any effects from it, like, directly impacting, you've just dumped 1.14×10¹⁸ kg of wood pulp onto Earth. Aside from any unpleasant effects from chemical additives, or blotting out the Sun's light to plants and causing biological collapse, I imagine that there could be some other unpleasant effects:
Earth's atmosphere is mostly nitrogen and oxygen, and by volume about 20% oxygen, so we'll back-of-the-napkin it and say that about 20% of the mass is oxygen. So about 1×10¹⁸ kg of oxygen, fairly close to the mass of the cardboard cutout.
That cardboard is presumably flammable.
So if this vast expanse of cardboard ignites --- like, from heat produced by falling through Earth's atmosphere, falling on any open flame in the world, or whatever, but seems like a pretty safe bet that something will touch it off --- I'd assume that a considerable portion of Earth's oxygen supply would be converted to water and carbon dioxide in the resulting combustion reaction. Even aside from the global wildfire itself, that seems like it'd be pretty bad news for humans.
The Sun's radius is ~696,000,000 meters, so the surface area of a perfectly circular cutout would be 1.5218e18 square meters. An article I found says that cardboard used for packing is about 0.35-0.4 kg per square meter, so taking an average of 0.375kg/m^2 gives a total of 5.7069e17kg. This is about the same mass as 40% of all water on Earth.
(I may mis-recall all the details because this was some years back.) There was girl who took a Danny Devito cutout as her "date" to highschool prom and posted the professional prom photo online, and the story got big enough that he heard about it. He was so amused by it that he brought a custom made cardboard cutout of her from the prom picture to the It's Always Sunny in Philadelphia set to take a photo with.
The Death Star. Not only would it be cool and intimidating, but I think it could be leveraged to reduce global warming by shading the world for about 12 minutes every day. (I attempted math to come up with 12 minutes. I wouldn't trust that figure, but it's all hypothetical anyway so the amount of time really doesn't matter.)