How much is the noise of a multiplexer for analog inputs?
You can calculate the thermal (Johnson-Nyquist) noise of the switch easily. That's broadband noise. 

\$v_{n}  = \sqrt{ 4 k_B T R \Delta f }\$   

where T is the temperature in Kelvin, R is the resistance, \$k_B\$ is Boltzman's constant, and \$\Delta f\$ is the bandwidth. 

Leakage current could cause some disturbance, especially to DC levels of your signal, you'll have to work that out from the impedances of both the switch and the rest of your circuit, and datasheet leakage current guarantees. 

For much more detail, you can refer to this paper.  

There might be a slight thermocouple voltage, but likely insignificant since they don't get warm, so the thermal gradients tend to be minimal. 

CMOS multiplexers do not have offset, hysteresis, shot noise (except a tiny bit from the body diode leakage) or other bad characteristics, just leakage and some (nonlinear) resistance that can cause distortion if care is not taken, but since your question is about noise I'll ignore that. 

Generally the 4052 is better than advertised in leakage, since it's a cheap part it's not tested to very low leakage levels. It has quite a large internal resistance, especially when operated at lower voltages. 

Better guaranteed performance is available (at a price) by using analog multiplexers such as the ADG series.