This is overblown and it hasn't even started. It won't (well, very, very unlikely to) bring down the government.
There you go, that's the long story short.
But, it'll be a PITA for the government and may induce some Australian market volatility (as if ...)
The dets (bolding is mine) ...
- Opposition is considering a High Court challenge against a Coalition MP's right to stay in Federal Parliament, in a move that would put the Turnbull government's slim majority in peril.
- If Dr Gillespie was disqualified it would spark a byelection in his seat ... potentially bringing down the government.
- the Nationals would be well-placed to retain the seat
More here: Turnbull government at threat as Labor plans High Court challenge
