No, MissingNo. affects saves, but only the area containing Hall of Fame data (I refer to RB MissingNo here, Yellow one is a very different issue).
When encountering any Pokemon in the game, save bank with Hall of Fame data is opened. The reason being that the area just before Hall of Fame used for decoding sprite data - there isn't enough RAM to store all information, so it stores some of it on on-cartridge memory. For this reason, Pokemon games with physically not working save RAM don't display any Pokemon sprites or display them incorrectly.
However, MissingNo. sprite goes outside of sprite buffer, and overwrites Hall of Fame data. The sprite is constant (read from ROM, not RAM), so it always corrupts the same way, and due to that there is no risk of it doing something unexpected.
If you won't view Hall of Fame (on PC), MissingNo. should be completely safe. However, Hall of Fame has garbage data, which means it can load other glitch Pokemon that can corrupt information in RAM - but this is likely to either do nothing or crash the game with Hall of Fame data open for saving so there is no risk of losing the save. It's possible to repair Hall of Fame data by beating Elite Four 50 times without seeing MissingNo.'s sprite.
MissingNo. also sets highest bit of sixth item in inventory when seen. This increases its count by 128 unless you had 128 or more of it already. This is because the code to handle seen Pokemon considers Pokemon 0 to be 256, and then goes beyond area of RAM reserved for seen Pokemon, accessing the item data.
Garbled sprites after seeing MissingNo. are caused by leaving the flip flag set after decoding the sprite. This normally should never happen, but it did, because MissingNo.'s sprite is not intentional. Seeing a sprite in Pokedex should reset the state of this flag.