Most probably the edges of your UV-map faces are to close to the edge of the corresponding texture imagery underneath.
As you can see in the four zoomed-in UV Editor windows, the UV-map is neatly within the borders of the blue square in the texture. But because the texture is blurred (or smoothed), the orange background color bleeds into the UV-mapped area. This causes the plane in the 3D Viewport to have blurry orange edges.
The texture is blurred because the top entry on the Image Texture node is set to Linear. Note how a single green pixel in the bottom right corner is blurred.
To fix it you can simply move the edges that suffer from the color bleeding away from the border and away from that color.
Alternatively, if you set the top entry on the Image Texture node to Closest, it will also solve the color bleeding problem, but you will get a pixel perfect display of the texture on your material and corresponding model, as commonly seen in Minecraft models and textures. (The single green pixel now looks like a pixel)