Frequency response and sampling theorem for triangular function
You're right that a triangular function is not band-limited and that it can't be sampled without introducing aliasing. However, this doesn't mean that linear interpolation can't be useful. The usefulness of linear interpolation depends on the data to be interpolated and on the desired interpolation factor. If the data have a low pass character, i.e., if they are already sufficiently oversampled, then linear interpolation can be good enough if the interpolation factor is not too large.
The obvious advantage of (bi-)linear interpolation is its simplicity and computational efficiency. In image processing it performs better than the common nearest-neighbor interpolation. However, if there are no real-time requirements and if computational efficiency is no big issue, then the standard interpolation method in image processing is the more complex bicubic interpolation.