Texture Filtering shows lines on models

[Mini--Joe]

If the texture filtering is set to anything other than "none", the textures on my model do this:




It appears to be showing more of the model's texture than it should. It seems to happen when viewing from certain angles and from far away. I am new to modelling and UV mapping, and I don't understand what is doing this.
Does anyone know how to stop this from happening?

[Hipshot]

Paste the texture, I just assume there's a white pixel row or similar outside the "dark" uvmapped area? This is not uncommon, happens in a lot of games, even modern, you just don't see it there because of the high resolution.

Like this, the white you see there might be the "face" part of the texture.

[Cherno]

Since texture filtering interpolates the pixels, there may be cases where the UV map gets "bleeding" from pixels of an unwanted adjacent color. The only thing to avoid this is to make sure that the UV mapping has extras space of at least one pixel around each area.