1) Every measurement is an interaction:

To gain information about a system, i.e. measure it, we must interact with it. For example we could perform a measurement by touch. Classically two objects interact by applying a force to each other. This then leads us to the conclusion that any reference frame which is measuring is non-inertial, and in classical mechanics (as well as in relativity) where we are performing continuous measurements, this will have a serious effect on our theory; obviously depending on the strength of the "force of measurement".

2) A measuring device measures itself:

For this point consider only the measurement of position. Space is relative; this means our measuring device can only measure the position of an object relative to itself. For this to be the case our measuring device will “know” that its position relative to itself is zero (we are here assuming that our idealized measuring device has zero extent). You may consider this point trivial, but you are fortunate enough to be endowed with a consciousness; surely we cannot assume every measuring device has a consciousness, but if our measuring device measures its own position this problem is resolved.

Now though we have to deal with complications arising from this self measurement. I cannot think of any classical problems (I would be glad if someone can come up with some) but they can arise in the case of QM, especially with continuous self measurement. Basically if our quantum mechanical continuous measuring devices self-measures then its wave function will always be (in its reference frame) proportional to δ(x), and this will lead to problems if we have other types of measuring devices, i.e. momentum measuring device.