Wer auf die Jagd geht, verliert seine Stelle.

Italian Translation

Chi va a Roma perde la poltrona.

The right translation should be “Chi va a caccia, perde il posto”

I am not sure about that, but isn’t the Italian sentence ““Chi va a Roma perde la poltrona” a political innuendo to the children’s play “Sedie musicali”, in English “Musical chairs” or Trip to Jerusalem” and in German “Reise nach Rom”.
I think it is used when a new government is formed.

Nevertheless you are right, the German sentence does not fit the Italian one.
It’s probably a misunderstood coupling via a third language (French?).

It has nothing to do with a ‘new government’. It is just a way to say that when you leave a place where you usually had your business you’ll take the risk to loose your position (can be politics but even other stuffs…,)

Sorry, my memory misled me.
I mixed it up with “di valzer delle poltrone”.
See Sedie musicali - Wikipedia