Last year we had a viewer who had a burn hole in a counter top from a hot pot. We showed how you can use the router to cut out the burn and create a recessed hole for a tile or a cutting board. See "The Plunging Router meets the counter top burn" .
This year we got the same question again but this time burns from incense cones, so wanted to take the process a bit further. As you router out that hole, make sure that the depth is such as to assure that the tile will stand a bit above the counter. This will help avoid burning again if you put a hot pot on the tile, it will float slightly above the counter. I recommend making the depth match just where the tile sloped down to the side where it would normally be grouted.
Now, the most important part of this step is to protect this cut in the counter from water. So use silicone caulking as a glue, and put it liberally under the tile, but especially heavily all around the edge. When you put in the tile, it must ooze up in such as way as to completely seal the joint and prevent any water from getting under the laminate on the counter. Don't wipe it off but let it set and then remove it with a razor blade.