Tameshigiri is the action of using a sword to cut objects (tamesu in Japanese means "try out", while giri means "cut").
Iaido is the only martial art where one practices with a sword in order to "polish" ones spirit/heart (kokoro). Tameshigiri is forbidden by the Zen Nihon Iaido Renmei, the only specialist iaido organization in Japan.
There are several reasons for not doing tameshigiri.
Firstly, one has to consider that the Japanese sword is not a tool, irrespective of whether the blade is live or not. In Japan, the Japanese sword has held a very important, and sometimes mystical, position in society. It is still one of the 3 treasures of the Imperial family, along with the mirror and orb. Thus, a sword is thought to have sacred properties.
Taking this into consideration, the act of using a sword to cut something is akin to someone using a Catholic cross to bash a tree.
Furthermore, swordsmiths put their heart and soul into making Japanese swords great works of art. Therefore, using a sword to cut something is a most disrespectful act towards the swordsmith and the sword, which samurai thought of as their soul.