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:0704.3878) 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.