I took up my camera by the lens when the camera was attached to a tripod. Will it cause any damage?
No, at least not on the camera. The lens mount of the camera is designed to support the weight of the camera (and some accessories). When used with big heavy lenses (anything bigger than 200mm) the lens supports the camera and not the other way around (a tripod is attached to the lens).
Damage to the lens is quite unlikely, too.
Maybe it will not cause damage maybe it will, but if is a risky way of picking it up why risk it?
It made you come here and ask this, so you did not feel comfortable in doing it. So just pick it up by the camera instead.
Strange question to be asked to be honest. You felt it was not right, the same way we would probably feel. Changing the way you pick up the camera is not a life changer so, just pick it up in a way that does not make you question yourself if you are doing things right or not.
And why would you prefer to pick it up by the lens anyway? If you were looking for: "Its fine man I do it all the time".
and even if anyone does this, each camera can be more fragile in its own way. so the obvious answer would always be: just pick it up normally.