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) 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.
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 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.