Why do objects feel heavier when held with an extended arm than with a bent arm? I do realise that this is due to torque and that torque is at maximum when the angle between the direction of the torque and the force acting on the object is 90 degrees. I would like to know if it is the angle between the forearm and the elbow that causes the situation I described in the question or whether if it's due to something else. 
 A: FGSUZ and Martin Ueding answers already cover the basics of torque. But a rough diagram might be helpful. Answering directly the question

it's due to something else

It's due to reduced lever arm $L$, so yes, it's due to the angle between the forearm and the elbow, but only to the extent to which this angle leads to the reduction of $L$ from $L_\mathrm{extended}$ to $L_\mathrm{bent}$.

A: I'd also say that this is due to torque. It is not the angle at the elbow. Take some weight in your hand and let it hand straight down from your shoulder. Then pull it up to your armpit. Holding it with an extended arm will be easiest because you only need to apply force in your hand.
When you have the weight in front of you, same height as the shoulder, stretched arm, it will feel rather heavy. You can rotate your elbow downwards such that it locks. Now you do not need any force around your elbow joint, but your shoulder has to counter all the torque that the weight exerts on you via your arm. The muscles are attached via tendons rather close to the bones, therefore they don't have much leverage. Therefore they need to exert a lot of force to keep the weight up.
Moving the weight closer to your body but keeping its height by closing your elbow will lessen the torque on the shoulder and therefore reduce the force that the muscles there have to apply.
A: You should always choose to calculate the torque with respect to the axis of rotation in each case. 
In one of the cases, that point is your elbow and so the distance is elbow-hand, while in the other case it is the distance shoulder-hand=whoele arm.
Evidently a longer distance makes greater torque, and so more propensity to turn around. 
Evidently, the rest of your body would turn around in the opposite direction if there weren't other forces. The thing is that your body makes forces in the appropiate points so that you stay stood up, but the nature of these forces is not relevant here. Just care about the torque produced by the weight in your hand.
