In Diablo 3 you can rune each skill once you get to an appropriate level. I notice that quite often the first rune you learn is "improved damage". Is there any benefit in not runing a skill, or should you always have some rune applied if you can use one?
For some skills, the runes really change the behavior and are not a flat-out bonus, so for these skills then yes, if that's the only rune available then there's a reason not to rune it. For example, see my question regarding Does Impending Doom decreases Exploding Palm's DPS?
A case where it's the only rune available is, for example, with Elemental Arrow. The first rune decreases the projectile's speed, so if you don't want that then you'll have to wait until the next rune, 5 levels later. So for 5 levels you'll have a reason not to rune the skill.
But the overwhelming majority of runes do not change the inherent skill behavior, only add to it, and they are a clear upgrade.
No, if you have all runes unlocked for a skill, always at least one of them (usually more) will give some form of flat advantage, without any disadvantages (even if other runes to the same skill might give situational disadvantages).
E.g. Exploding Palm, which has a lot of runes with situational advantages, the first rune still has a flat advantage.
However, while leveling up, there are some skills where the use of any runes is situational if you only have a few unlocked, because the original skill without runes might indeed be better in certain situations.
E.g. the Witch Doctor's Poison Dart ability does 100% weapon damage and 60% as damage over 3 seconds. The first rune instead shoots 3 darts at 60% damage. If you only have the first rune of all of them, then against single enemies (e.g. bosses) the base ability might be better, because you will usually only hit with one dart of the three, effectively lowering the ability damage to 60%, where it could be 100%.