Note that the moments of the lognormal are $ {E} [X^{n}]=e^{n\mu +{\frac {n^{2}\sigma ^{2}}{2}}}$, and if you know you're dealing with a lognormal even specifying the first two moments ($n=1,2$) will identify a particular lognormal.
The actual situation intended by "the lognormal distribution is not uniquely determined by its moments" is that for every lognormal distribution, there's another not-lognormal distribution (in fact an infinity of them) with the same set of moments.
Specifically, there's a family of distributions indexed by two parameters $0<\epsilon<1$ and integer $k>0$ (that includes the lognormal as a limiting case, with $\epsilon=0$ for any $k$) which has a moment sequence that doesn't alter with $\epsilon$ or $k$.
Stated most simply, if $f$ is a standard lognormal density ($\mu=0$, $\sigma=1$) and $g(x)=f(x)\cdot (1+\epsilon\sin[2\pi k\log(x)])$ then the contribution of $g-f$ to the $n$th moment is $0$ for each $n=1,2,...$.
(By change of variable the result extends to the whole lognormal family)
In fact Chris Heyde demonstrates this[1] for the (slightly) more general case of a shifted lognormal (three parameter lognormal; Heyde refers to it as a "general lognormal"). Again, this further extension of both $f$ and $g$ to cover the three parameter lognormal case simply involves another change of variable.
[1] Heyde, CC. (1963),
"On a property of the lognormal distribution",
Journal of the Royal Statistical Society, Series B, 25 (2): 392–393
(pdf here: http://link.springer.com/chapter/10.1007%2F978-1-4419-5823-5_6 )
