Yes, indeed any measurement where you learn something will disturb the state. This is also known as "no information without disturbance". Recently, there has been a lot of flurry of activity trying to pin this down quantitatively.
As you said, this is not what the Heisenberg relation you write down is about, but the effect still certainly exists. Here is a number of questions you could ask (let's stick to position and momentum):
- Given a quantum state (i.e. a preparation procedure, which is to say a number of steps you do in a lab), what are the fundamental limits on the variance of position and momentum if I measure them (separately)?
- Given a quantum state, what happens if I measure the position of the state and then the momentum of the state and compare the momentum distribution with the distribution I obtain when I only measure the state?
- Given a quantum state, what happens if I measure the position only very coarsly and then measure the momentum? Will anything change from the previous measurement?
- What is the best way to jointly measure position and momentum? I.e. what measurement can I do that outputs two values every time I measure, one for position the other for momentum. The "best" meaning that the distributions are as close as possible to the distributions where I just measure position or just momentum.
The first is as far as I understand it pretty much what the Heisenberg uncertainty principle describes. The second and third scenario describe a qualitative and quantitative analysis of how much a measurement actually "disturbs" a state. The fourth asks about doing the measurement at the same time (something that you also often hear in connection with the HUP).
As I understand it (but I am by no means an expert), people are getting a much better understanding of the scenarios two to four in various flavour. Note for example that you always have to compare probability distributions for which there are many ways to do it.
Let me close by giving one link to a paper that tries to distinguish a number of different relations (operationally). It might not be the best to start with, but it seems reasonable and if you are interested, you can get a lot from there, since it is a pretty recent paper: http://arxiv.org/abs/1402.6711