Here is another interesting case where the image of the norm map can be written down explicitly.  Let $F$ be a finite extension of $\mathbf{Q}_p$ containing a primitive $p$-th root $\zeta$ of $1$, and denote the filtration on the multiplicative group $F^\times$ by 
$$
\ldots\subset U_2\subset U_1\subset\mathfrak{o}^\times\subset F^\times.
$$
We thus get the extensions $L_n=F(\root p\of{U_n})$.  It can be checked that $L_{pe_1+1}=F$, where $e_1$ is the absolute ramification index of $F$ divided by $p-1$, and that $L_{pe_1}$ is the unramified degree-$p$ extension of $F$, so the image of the norm map $L_{pe_1}^\times\to F^\times$ is $\mathfrak{o}^\times F^{\times p}$.

What is the image of the norm map $L_{n}^\times\to F^\times$ for other $n$ ?  Local class field theory and a certain orthogonality relation for the Kummer pairing (see for example the last section of [arXiv:0711.3878][1]) can be used to answer this question.  Basically, for $n\in[1,pe_1]$, $N(L_n)=U_mF^{\times p}$, where $m=pe_1+1-n$.

There are similar results for elementary abelian $p$-extensions of finite extensions of $\mathbf{F}_p((t))$.  See for example the last section of [arXiv:0909.2541][2].

These two papers have appeared in *J. Ramanujan Math. Soc.* **25** (2010), no. 1, 25–80, 
and **25** (2010), no. 4, 393–417.

There are other instances where the image of the norm map can be computed explicitly.  This happens for the cyclotomic extension $K_m$ of $\mathbf{Q}_p$ obtained by adjoining $\root{p^m}\of1$.  It is can be shown that $p\in N(K_m^\times)$, and that the image of the units of $K_m$ under the norm map down to $\mathbf{Q}_p$ is $1+p^m\mathbf{Z}_p$.  See for example Artin,
*Algebraic numbers and algebraic functions*, p. 208, or Neukirch,
*Class Field Theory*, p. 45.

  [1]: http://arxiv.org/abs/0711.3878
  [2]: http://arxiv.org/abs/0909.2541