I am installing ceramic tile in my laundry room , whick also includes my oil furnace ,should I tile around and under my furnace or leave an area for inspections and maint.so as not to stain tile and grout.

I would go around it... and store extra tiles to do it under it as well in the future if you ever have to replace it (not storing it UNDER the furnace obviously). If you ever replace it odds are the footprint won't be the same. It gives a more finished look and costs a minimal amount more.

I agree, tile around it. You aren't going to see under the furnace and you sure won't just lift it up to tile.