No. There are a large number of private subway and train companies in Tokyo and other cities. In the countryside you will have less of an issue, but in central Tokyo you will only be able to use a small number of lines. Having said that, you can get to most of the major spots in Tokyo using JR, it just may take a little longer than using the subway. Even if you do decide to use the subway occasionally, it's only usually 170 yen (less than US$1.50) for a trip so it's not going to add up to a lot. The JR Pass is certainly good value if you are using the bullet train to travel between the major cities in Japan. If you are just staying in Tokyo you don't need it. Here's a map of the lines you can use: http://www.jrpass.com/map/
Actually subway is more often 200 yen, plus if you have to change the trains, the sum may add up unexpectedly.
As Andrew said and as they say on japan-guide.com,
The Japan Rail Pass (also commonly called JR Pass) is a very cost effective rail pass for long distance train travel in Japan.