MSQ progress by itself will not prevent you from leveling up. Howver, unless you have the Heavensward expansion you will not be able to level beyond level 50. (Both the Heavensward and Stormblood expansions are included if you buy Shadowbringers - the latest expansion)

If you have the Road to 70 buff you will gain EXP very quickly, so using one or two extra classes to absorb all that extra EXP is a great idea - otherwise you will just outlevel quests very quickly.