Renting a car from Milan to Switzerland

There are few things you need to be mindful of:

• Are you allowed to take the car abroad at all? In Western Europe (but not even the whole EU) it's usually possible but not always and not everywhere, you need to check the conditions.
• Are you insured in Switzerland with an Italian car? Same thing, usually it's included but do check (at least I know Italy is typically covered for cars hired in Switzerland, I assume it should be the same the other way around).
• Can you return the car in another country? It seems you don't want to come back to Italy at the end of your journey so you need to return the car in Switzerland. I don't know for sure but I assume local or independent rental company will not offer this service. You would therefore need to look at big international brands, which will simplify your search. It should not be too difficult to track down the websites and check the conditions. There will obviously be a (potentially steep) charge for this (see also Inter-country one-way car rental in Europe?).
• Motorways and transport through the Alps. In Switzerland you need a vignette to use the motorways. I am pretty sure that an Italian car will not have it so it's not “inclusive” but you can always ask (maybe they even have a Swiss car they need to get back to Switzerland, you never know). If not, it's just something you can buy at the border. Motorways aren't free in Italy either.

In short, driving into Switzerland should be possible without any extra charge as far as the car hire company and insurance are concerned but you will almost surely need to pay for tolls yourself. Dropping the car there will be very expensive in any case. Do read the fine print or ask them about all this beforehand.

One attractive alternative would be to look for a car that needs to be relocated but you obviously need to be lucky and flexible to find something that fits your needs. As of writing this, Hertz seems to have a car in Milan Malpensa that needs to be brought back to Switzerland so it is indeed possible to find something on this route. Considering the fact that Milan is a major business center and quite close to Switzerland, the odds of finding such a car when you need one by calling all major car rental companies might not be too bad.

To drive in Switzerland you need the road pass (motorway vignette) which is valid for 1 year. If you will rent the car without it you should buy it to enter Switzerland. Usually you can find it for sale in customs offices at the Swiss border. The driving without this pass is the subject of fine. Moreover probably border officers control it also.
I suppose you also do not want to make the present to the rental car company so you can try the trick not to apply the pass on the window but to put it on to be easy removable and take with you out of car for next car rental and driving to Switzerland.
The actual cost should be nearly 40 francs i.e. ~36 EUR