What Is the Difference Between MRF and Total Variation in Noise Removal?
These are two different concepts that you talk about. First, MRF gives you a framework to do discrete optimization of problems, which respect the Markovian property, that is a pixel is conditioned only on the neighboring ones (roughly stated). Typical applications include binary or multi-class labeling problems. Total variation on the other hand, is generally used as a regularization by adding the integral of the absolute gradient of the signal/image to the energy functional. This helps to neglect irrelevant detail and focus on important ones. 

You cannot say one is better than the other, as they are not exactly contradictory things. It depends on the application and the energy function you use in the MRFs. 

An Introduction to Total Variation for Image Analysis is a good total variation tutorial to begin with. Also, Chambolle provides a link between total variation and binary MRFs, leading to new algorithms in the work entitled Total Variation Minimization and a Class of Binary MRF Models:


  Total Variation Minimization and a Class of Binary MRF Models, Antonin Chambolle, 2005