Not getting the correct asymptotic behaviour when sending a small parameter to zero

You could try using AsymptoticSolve instead:

AsymptoticSolve[
    I η - (2 Pi^2/β^2) z^2 + (2 I Pi^2 u/β^2) z^3 + z^4 == 0,
    z,
    η->0
] //Normal //TeXForm

$\left\{\left\{z\to -\frac{\sqrt{i \beta ^2} \sqrt{\eta }}{\sqrt{2} \pi }\right\},\left\{z\to \frac{\sqrt{i \beta ^2} \sqrt{\eta }}{\sqrt{2} \pi }\right\},\left\{z\to \frac{\pi \left(-\sqrt{2 \beta ^2-\pi ^2 u^2}-i \pi u\right)}{\beta ^2}-\frac{\beta ^6 \eta }{4 \pi ^3 \left(-i \pi ^2 u^2 \sqrt{2 \beta ^2-\pi ^2 u^2}+i \beta ^2 \sqrt{2 \beta ^2-\pi ^2 u^2}+\pi ^3 u^3-2 \pi \beta ^2 u\right)}\right\},\left\{z\to \frac{\pi \left(\sqrt{2 \beta ^2-\pi ^2 u^2}-i \pi u\right)}{\beta ^2}-\frac{\beta ^6 \eta }{4 \pi ^3 \left(i \pi ^2 u^2 \sqrt{2 \beta ^2-\pi ^2 u^2}-i \beta ^2 \sqrt{2 \beta ^2-\pi ^2 u^2}+\pi ^3 u^3-2 \pi \beta ^2 u\right)}\right\}\right\}$

Execute

s1 = Solve[-2*(Pi^2/\[Beta]^2)*z^2 + 2*I*((Pi^2*u)/\[Beta]^2)*z^3 + 
z^4 == 0, z]; s2 = 
Solve[I*\[Eta] - 2*(Pi^2/\[Beta]^2)*z^2 + 
2*I*((Pi^2*u)/\[Beta]^2)*z^3 + z^4 == 0, z]; 

and then compare

s1 /. {\[Beta] -> 0.7, u -> \[Pi]}

with

s2 /. {\[Eta] -> 0, \[Beta] -> 0.7, u -> \[Pi]}

to see that both equations numerically have the same solutions.