Does SNR improve at higher sampling rates for a low pass filtered signal?
I see quite a few misstatements and possible misconceptions both in the answers and in the question, so let’s break down what is meant by “noise.”


Analog noise within the same bandwidth as the signal. 
Analog noise outside of the bandwidth of the signal. 
Quantization noise introduced by the ADC. 


I interpreted your question as implying option 1. Increasing the sampling rate will do absolutely nothing regarding that noise. No ifs or buts. That noise is there to stay.

Other answers only apply to options 2 &amp; 3. 

If your analog filter is not steep enough and you still have strong noise components above the bandwidth of the signal (option 2), particularly near the sampling rate and its multiples, then increasing the sampling rate and using an appropriate digital filter will allow you to (1) relax your analog filter requirements and (2) completely remove that out of band noise. 

Regarding option 3, that’s the basic operating principle that makes sigma-delta converters possible. Increasing sampling rate, using an appropriate digital filter at a higher resolution, and decimating the output of the filter to the needed rate, allows you to increase the resolution of your samples.

You can gain 1/2 bit per each doubling of sampling rate, as long as your signal is large enough to remain uncorrelated to the sampling noise. Interestingly, this is one place where out of band noise works in your favor, as it breaks the sampling noise correlation to the signal. I.e., it serves the function of dithering (or, as physicists call it, stochastic resonance). 

For slow enough signals this is a great advantage, using this technique I have obtained ~20 bits of resolution out of 12 bit converters by purposefully designing the analog anti-alias filter to introduce several bits of out-of-band random noise and oversampling by a factor of 4096.