In this paper, IsoGeometric Analysis is applied to a transient nonlinear heat conduction problem using a predictor-corrector scheme. To obtain a better approximation of the solution, we propose the k-refinement method coupled to a Matrix-Free Weighted Quadrature approach which speeds up matrix-vector operations and also reduces memory consumption. The main goal of this work is to introduce a preconditioning strategy to improve the convergence rate of the iterative solver. This method is based on Fast Diagonalization technique which exploits the tensor structure of basis functions. We show that our preconditioner is inexpensive, easy to code and as effective as other more sophisticated methods. It is also robust with respect to polynomial degree, mesh size and time-integration scheme parameters. Numerical experiments for different geometries and materials are presented, which support our theoretical results.