The existing answer is correct in that Missingno is generally harmless. However, it does technically harm your save file -- but not in a way that is particularly significant or that inhibits gameplay.
Certain circumstances, generally involving the use of Missingno in battle, will cause Pokemon sprites in the game to become scrambled or inverted. This is only a temporary effect, though; looking at the summary of a normal Pokemon will reverse these effects.
Missingno will, however, cause some permanent damage to the save file. It corrupts the game's Hall of Fame entries, causing graphics to become scrambled and sounds to work improperly. However, this is the only damage that the glitch will have on the save file forever, and considering that the Hall of Fame is not necessary for any gameplay elements, it is generally considered a worthy sacrifice for having access to the item duplication glitch.
Missingno is frequently blamed for some particularly impeding effects on the game such as the internal battery running dry or for rendering save files unusable. However, these are merely rumours; the glitch itself is not to blame for such effects. If you'd like to know more about Missingno, the article linked here is a very good read.