When you do \addplot [mark=none] ..., the effect is that mark=none completely overwrites any style given by the active cycle list. That means in turn that the default line style of TikZ is used, which is just a black line. You want \addplot +[mark=none], where the + indicates that mark=none is appended to the style given by the cycle list.
That is just part of the answer though. In the code you show you have only defined a color map, you haven't actually used it. Add colormap name=bright to the axis options to activate it. But the colormap doesn't directly influence the style of line plots, in order to pick colours from the colormap for line plots, add for example cycle list={[of colormap]} to the axis options. Because the bright map you defined has eight colours, this cycle list will get eight colours as well.
If you want a given number of equally spaced samples from the colormap, you can use e.g. cycle list={[samples of colormap={20} of bright]} to get 20 samples (you don't have to specify colormap=bright in this case). You can also get non-equally spaced samples, by using e.g. cycle list={[colors of colormap={0,300,600,1000}]}. You supply a list of values between 0 and 1000, where 0 is the start of the map, and 1000 is the end.
All this colormap vs. cycle list stuff is described in the pgfplots manual, at the end of section 4.7.7 Cycle Lists – Options Controlling Line Styles, page 219. (In the manual for version 1.16 of pgfplots, dated 2018-3-28.

\documentclass{scrartcl}
\usepackage{pgfplots}
\usepgfplotslibrary{colormaps}
\pgfplotsset{compat=1.9}
\pgfplotsset{
colormap={bright}{rgb255=(0,0,0) rgb255=(78,3,100) rgb255=(2,74,255)
rgb255=(255,21,181) rgb255=(255,113,26) rgb255=(147,213,114) rgb255=(230,255,0)
rgb255=(255,255,255)}
}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
axis y line*=left,
axis x line*=top,
ylabel near ticks,
xlabel near ticks,
xlabel=Current in A,
ylabel=Voltage in V,
every x tick label/.append style = {font=\footnotesize},
every y tick label/.append style = {font=\footnotesize},
colormap name=bright, % activate the defined colormap
cycle list={[of colormap]},
every axis plot/.append style={mark=none,ultra thick} % set options for all plots
]
\addplot {x};
\addplot {x+1};
\addplot {x+2};
\addplot {x+3};
\addplot {x+4};
\addplot {x+5};
\addplot {x+6};
\addplot {x+7}; % white, so not visible
\end{axis}
\end{tikzpicture}
\end{figure}
\end{document}