Does the CMB signal get weaker over time? If the universe is infinite or flat, then this isn't true (I guess).
But if the universe is finite, then as it expands wouldn't the CMB signal weaker at any given point over time?
 A: If we define a scale factor $a$ for the universe (could be the distance between two galaxies), then this scale factor will change with time. This is also true in flat or infinite universes, so long as the Hubble parameter is $>0$ (i.e. the universe is expanding).
The energy density contained in the cosmic microwave background will scale as the energy of the CMB photons divided by the volume they occupy.
$$ u_{\nu} \propto  \frac{h\nu}{a(t)^3},$$
where of course the volume of a chunk of space increases as $a^3$. This assumes that the number of photons is a conserved quantity.
At the same time we know that photons are being redshfted - their wavelengths are stretching in exactly the same way as $a$. That is $\lambda \propto a$. But for photons $\nu = c/\lambda$, so $\nu \propto a^{-1}$.
Overall then, we see that the energy density of the CMB decreases as $a^{-4}$. Thus in an expanding universe, as $a$ increases, the the energy density of the CMB decreases a lot and the "CMB signal" that you refer to will indeed get weaker.
Another effect is that the temperature of the CMB will decrease. It is currently at about 2.7 K, but for a blackbody spectrum, the temperature follows Wien's law such that $T \propto \lambda_{\rm peak}^{-1}$, where $\lambda_{\rm peak}$ is the peak wavelength of the blackbody spectrum. But we have already seen that wavelength is proportional to the scale factor, so as $a$ increases, $\lambda_{\rm peak}$ increases and $T$ decreases as $a^{-1}$. See also CMBR temperature over time?
The relevant timescale on which this occurs
$$ \frac{du_{\nu}}{dt} \propto -4a^{-5} \frac{da}{dt}$$
$$ \frac{d u_{\nu}/dt}{u_{\nu}} = -4 \frac{da/dt}{a}$$
But the ratio of $da/dt$ to $a$ is the Hubble parameter, currently thought to be about 70 km/s per Mpc (or about $2.3\times10^{-18}$ s$^{-1}$).
$$\frac{d u_{\nu}/dt}{u_{\nu}} = -4 H_0$$
and the timescale to measure a fractional change $\Delta u_{\nu}/u_{\nu}$ is therefore
$$ \Delta t \simeq  \frac{-1}{4 H_0} \frac{\Delta u_{\nu}}{u_{\nu}}$$
So a 1 percent change in the energy density of the CMB will occur over 35 million years.
