Counterexamples for uniqueness of viscosity solutions Recall a classical comparison result for viscosity solutions:

Let $H:[0,T]\times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ 
  
  
*
  
*continuous
  
  
  satisfying 

  
*$\vert H(t,x,p)-H(t,x,q) \vert \le C \vert p-q \vert$ 
  
*$\vert H(t,x,p) - H(s,y,p)\vert \le C(\vert t-s \vert+ \vert x-y \vert)(1+\vert p \vert)$ 
for some $C>0$. Let $\underline{u},\overline{u}$ bounded and uniformly continuous be sub- and a supersolutions of the Cauchy problem for $$\frac{\partial u}{\partial t}+H(t,x,Du) = 0.$$ If $\underline{u}(0,x) \le \overline{u}(0,x)$ for all $x \in \mathbb{R}^n$, then $\underline{u} \le \overline{u}$ in $[0,T]\times \mathbb{R}^n.$

Can you give counterexamples that show that the claim does not hold if we remove any one of assumptions 1 or 2 or remove parts of 3?

I assume you are interpreting viscosity solutions via Ishii's notion of solution when $H$ is discontinuous (where you replace $H$ by its upper and lower semicontinous envelopes in the super and subsolution properties, respectively). In this case consider the PDE (or ODE rather)
$$u'(x) = f(x)$$
where $f(x)=1$ for $x$ rational, and $f(x)=0$ for $x$ irrational. Comparison does not hold for this PDE (any linear function $u(x)=mx+b$ with $0 \leq m \leq 1$ is a viscosity solution in the Ishii sense. 
It is possible to prove uniqueness when $H$ is discontinuous, but the discontinuity has to be relatively mild (say, a jump discontinuity along a Lipschitz surface), and other conditions have to be placed on $H$. 
