One of our viewers has removed a stove and has an unused chimney going up the outside of the house. The top has been capped off, what to do with the bottom?
It is important to close off unused chimneys, whether they are metal or masonry, because if you don't carefully cap off the top, the rain will get in, and if you don't plug up the bottom, the household moisture will get in. In either case, unknown to you, the metal chimney will slowly rust and the masonry chimney will slowly decay.
There is one important detail for the top cap. It must be a complete rain shelter, not just the spark cap that is normally on a chimney, but at the same time it must not be sealed air tight. Usually unused chimneys are capped off with a bent sheet metal covering and perhaps a wire strap or weight on the top. The rain cannot get in, but it is not sealed air tight, which does allow any inside moisture to slowly get out.
The bottom is even easier. Simply cut a tight fitting plug out of foam insulation and block the hole on the bottom. You could caulk around the edge if you really don't think you will ever use this chimney again, or if you want to be able to easily rehabilitate it later, use the foam rubber weather-stripping tape to seal the edges. We do need it air tight on the bottom.
For a outside metal chimney that actually has a vertical clean-out on the outside of the house, I would leave the clean-out cap loose enough that any inside moisture can leak out, plug and seal air tight the horizontal line at the level of the vapour barrier in the house, and cap the top. 
But for a masonry chimney that has a clean-out in the basement, tape that clean-out air tight as well, because it can be a source of warm moist air getting inside the closed chimney. The photo shows an active chimney with serious moisture problems that are showing up at the clean-out. This can actually get worse when the chimney is no longer used and improperly closed off.