  We consider entropy conservative and dissipative discretizations of nonlinear
conservation laws with implicit time discretizations and investigate the
influence of iterative methods used to solve the arising nonlinear equations.
We show that Newton's method can turn an entropy dissipative scheme into an
anti-dissipative one, even when the iteration error is smaller than the time
integration error. We explore several remedies, of which the most performant is
a relaxation technique, originally designed to fix entropy errors in time
integration methods. Thus, relaxation works well in consort with iterative
solvers, provided that the iteration errors are on the order of the time
integration method. To corroborate our findings, we consider Burgers' equation
and nonlinear dispersive wave equations. We find that entropy conservation
results in more accurate numerical solutions than non-conservative schemes,
even when the tolerance is an order of magnitude larger.
