I'm trying to make an icosphere with a hexagonal pattern applied to it, but the image I'm using ends up getting completely warped and ripped along the faces. I don't know how to fix this, so please help.
There are two ways to approach this problem.
1) UV Unwrapping
There is a major problem with texturing a 3D sphere.
It is mathematically impossible to map a 2D texture onto a 3D sphere without distortion.
Here is a great Stack Exchange thread where such situation and possible solutions are discussed in detail.
2) Procedural textures
If your texture is simple, you might want to use procedural texturing, instead of an image.
This way you can get a seamless pattern without any distortion.
Here is an official Blender video about procedural textures.
If you'll need any clarification, let me know.
You might want to try the node setup shown below. Change the parameters in the mapping node to move the image around. This might not work, as it is possible that the problem is in the UV projection, but I've already solved problems like this using this node setup.
I hope this helps.