Properly installed drip edge should last the life of the roof but it can be damaged by falling limbs or other debris, which can bend it out of shape. If the shingles are still intact and in good shape, you can remove the drip edge by lifting the shingles slightly and prying up the nails with a pry bar. Install new drip edge with roofing cement on the roof decking unless you can lift the shingles enough to drive new nails.
If the drip edge is draining water behind the gutter instead of into it, replace it with a different style. Basic drip edge is metal bent 90 degrees, with a flat edge to go under the roofing and a downward edge over the fascia or facing board, with a lip at the bottom to push water away. A P-style will work better on an eave; this extends beyond the end of the roof and is turned back in a P shape before it is bent down the fascia. This provides an extra inch or so to direct water to a gutter.
Check to see whether the problem is with the roof decking or shingles. Decking must extend clear to the fascia without a gap; otherwise drip edge may tilt back down into that gap. Shingles need to slightly overlap the decking. If either decking or shingles are improperly installed and causing drip edge problems, you may have to remove the drip edge and patch the decking or shingles.
Many drip edge problems can be avoided with proper installation. Install drip edge first on the eave or bottom of a roof slope, underneath the roofing paper or felt, so any water that gets in above will drain off over the edging. Put drip edge on the rake or sloped edge of a roof over the roofing membrane so any water driven in by rain will drain down the roof without getting under the edging.
Drip edge also has to be correctly overlapped at corners and roof peaks. Make sure the rake or sloped edging goes over the eave or bottom edge, clear to the end. Nail it at the corner through both strips. Use tin snips to cut out a top section so there is a bottom piece to wrap around the corner to seal the corner. On peaks, run one strip to the top of the roof and overlap it with the strip on the other side to the top of the other slope. Nail through both strips.