This may be what you mean:
First, initialize (I use := instead of your = )
Clear[myfun, plotfn]
myfun[a1_, a2_, a3_, b_, \[Theta]_, \[Phi]_] := a1 + 2 a2 + 3 a3 + b Cos[\[Theta]] + (a1 Cos[\[Phi]] + a2 Sin[\[Phi]]) Sin[\[Theta]]
Then, a plot function which plots what you ask for if the condition that you define is satisfied. Otherwise, it prints the result of the condition
plotfn[a1_, a2_, a3_, b_] /; (Sqrt[a1^2 + a2^2 + a3^2] + Abs[b] <= 1) :=
ContourPlot[
myfun[a1, a2, a3,
b, \[Tau], \[Phi]], {\[Tau], -\[Pi], \[Pi]}, {\[Phi], -\[Pi], \
\[Pi]}];
plotfn[a1_, a2_, a3_, b_] :=
"Sqrt[\!\(\*SuperscriptBox[\(a1\), \
\(2\)]\)+\!\(\*SuperscriptBox[\(a2\), \
\(2\)]\)+\!\(\*SuperscriptBox[\(a3\), \(2\)]\)]+|b|" ->
N[Sqrt[a1^2 + a2^2 + a3^2] + Abs@b]
A Manipulate allows to vary a bit
Manipulate[
plotfn[\[Alpha]1, \[Alpha]2, \[Alpha]3, \[Beta]], {{\[Alpha]1, .1}, \
-1, 1}, {{\[Alpha]2, .1}, -1, 1}, {{\[Alpha]3, .1}, -1,
1}, {{\[Beta], .1}, -1, 1}]
edit: forgot the Abs around b in the condition