Consider the following state vector
$$\frac12 (|000\rangle + |011\rangle + |101\rangle + |110\rangle)$$
Can noise in measurements lead to a non-zero possibility of measuring the state $|111\rangle$ for example?
A state that was not originally in the state vector?
Or is the noise-effect limited to only changing the probabilities of states inside the state vector?
Measurement noise is typically considered a classical outcome that does not reflect the actual final state.
For your state vector, if you measure transversally - i.e. you measure, for example, in the Z basis each qubit individually -, then you can think of measurement noise as a possibility of each bit measurement flipping with a probability $p$. Thus it is possible to measure 111, while the true state will be one of the original terms in the superposition.
A bit more intuition on this: check out the Qiskit Textbook on discriminating between $|0\rangle$ and $|1\rangle$: https://learn.qiskit.org/course/quantum-hardware-pulses/calibrating-qubits-using-qiskit-pulse#zerovone - measurement results typically fall into one of the "clouds", and a classical classifier decides whether that point is 0 or 1 state - and it is not perfect. When it makes a mistake, it is considered a faulty measurement.
In QEC simulations, noisy measurement is usually modeled as a probabilistic bitflip on the measurement outcome.
