Resume
You can directly use the MCMC iterations for anything because the average value of your observable will asymptotically approach the true value (because you are after the burn-in).
However, bear in mind that the variance of this average is influenced by the correlations between samples. This means that if the samples are correlated, as is common in MCMC, storing every measurement will not bring any real advantage.
In theory, you should measure after N steps, where N is of the order of the autocorrelation time of the observable you are measuring.
Detailed explanation
Let's define some notation to formally answer your question. Let $x_t$ be the state of your MCMC simulation at time $t$, assumed much higher than the burn-in time. Let $f$ be the observable you want to measure.
For example, $x_t \in \mathbb{R}$, and $f=f_a(x)$: "1 if $x\in[a,a+\Delta]$, 0 else". $x_t$ is naturally being drawn from a distribution $P(x)$, which you do using MCMC.
In any sampling, you will always need to compute an average of an observable $f$, which you do using an estimator:
$$F = \frac{1}{N}\sum_{i=1}^N f(x_i)$$
We see that the average value of this estimator $\langle F\rangle$ (in respect to $P(x)$) is
$$\langle F \rangle = \frac{1}{N}\sum_{i=1}^N \langle f(x_i)\rangle = \langle f(x)\rangle$$
which is what you want to obtain.
The main concern is that when you compute the variance of this estimator, $\langle F^2 \rangle - \langle F \rangle^2$, you will obtain terms of the form
$$\sum_{i=1}^N\sum_{j=1}^N \langle f(x_i)f(x_j)\rangle$$
which do not cancel out if $x_t$ are correlated samples. Moreover, because you can write $j=i+\Delta$, you can write the above double sum as sum of the autocorrelation function of $f$, $R(\Delta)$
So, to recap:
If computationally it does not cost anything to store every measure, you can do it, but bear in mind that the variance can not be computed using the usual formula.
If it is computationally expensive to measure at each step of your MCMC, you have to find a way to estimate the cumulative of the autocorrelation time $\tau$ and perform measurements only every $\tau$. In this case, the measurements are independent and thus you can use the usual formula of the variance.