Why does Heisenberg picture formalism requires to rewrite operators in explicit covariance?
In the Schroedinger picture with QFT, the state is a function of a field configuration. It is possible to do QFT in the Schroedinger picture, but it is usually considered less intuitive. http://en.wikipedia.org/wiki/Schrödinger_functional
On the other hand in QFT the field operator is like a field on spacetime, except that it is "operator valued". Because of the analogy of the field operator to a classical field on spacetime, it is usually easier to write the time evolution of the operators, ie. it is easier to use the Heisenberg picture.
This is discussed by Tong in http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf.
But why the covariance formalism for Heisenberg Picture as I saw in some notes?
If you take a look at Eq 2.82 in the notes by David Tong linked above, you can see that the equation of motion for the field operator has the same form as the classical relativistic wave equation. The Heisenberg pictures and Schroedinger pictures are at equivalent the non-rigourous level, so it doesn't mean that the Schroedinger picture cannot be used as the basis of a relativistic field theory, it is simply that the Heisenberg picture is usually considered more intuitive.
Separate names with a comma.