Make router burn marks disappear
You thought you took every burn-proofing precaution. The wood thought otherwise. And judging by the scorch marks on that freshly routed edge, you lost. The only thing worse would be repeating your mistake, so let's figure out what went wrong before we fix the damage.
Care beats repair
First, check for the usual suspects:
1 Too much speed. Fast-turning large bits generate wood-searing heat. Back off the speed control to suit your bit according to these guidelines:
2 Heavy passes. Take off no more than 1⁄8 " of material with each pass—less on the final one.
3 Slo-o-o-ow pace. To avoid heat build-up, feed stock as quickly as possible without leaving chatter marks.
4 Dull bits. Replace or resharpen bits that have lost their edge.
This time for sure
Once you've identified the cause, make the symptoms—burn marks—go away. If they're only light scorches, as shown below, first try sanding them away starting with 100-grit abrasives up through 180 grit, or 220 grit if you'll add stain later.
Plane off a thin strip
For dark burn marks, rout a fresh profile that removes a thin layer and the burn marks along with it. First, using a hand plane, remove one or two curls from each edge that will ride along a router table fence or bit bearing, as shown below. If you don't have a hand plane, you can, instead, make a light shaving cut on your tablesaw, as shown in top photo.
Make paper-thin adjustments
If you're using a bit with no bearing on a router table and you can fudge the part size a hair smaller, move the fence back the thickness of a piece of paper, as shown below To do this, clamp stop blocks to the router table tight against the fence. Loosen and move the fence back enough to insert a piece of paper between the block and fence. Then snug the fence tight against the paper and stop blocks.
Now remove the blocks, and rerout the part at a faster feed rate. This will remove the burn marks with only minimal change in the part size.