No, put the razor down. If you wouldn’t shave over a sunburn or a scrape, you shouldn’t shave over a new tattoo. It can really impede healing and your tattoo won’t look as good in the future. Wait until your skin stops flaking and the area no longer feels raw and sensitive.
You should also avoid any harsh chemicals or hot wax. The only thing that you should be putting on your tattoo in the first couple weeks is unscented lotion.