Running wider tires in the front I currently run a full suspension bike with rather thick tires. I have however been thinking of changing the rear tire to something a bit thinner as the current tire I run makes the wheel very close to the linkage and frame. 

When there's even a bit of mud, the mud tends to stick to the wheel and the frame causing the rear wheel to completely jam. Is this a good idea or not recommended?
This is not a problem at all. 

What you see quite often is that riders use tires with a lighter tread and harder rubber compound on the rear of the bike to get the benefits of good grip on the front and less rolling resistance on the back. A common example combination would be to use the grippy Schwalbe Magic Mary on the front and the faster-rolling Hans Dampf on the back wheel. 
(I am not affiliated with Schwalbe, I just don't know the names of similar combinations of other manufacturers.)

Sometimes, this also includes using a slightly thinner tire on the back wheel. A small difference is fine, though I probably wouldn't go as far as using a heavy downhill tire on the front and a skinny gravel tire on the back ;-)