I have a multiple entry Schengen business visa issued by Switzerland which will expire in July. Can I travel to Switzerland next week for work and then travel again to France for work at the beginning of July for a week? Or do I have to apply for another visa at the French Consulate?
Your visa is generally valid for trips to France and other countries in the Schengen area. Provided it is a multiple-entry visa and you already used it for its main purpose, going to France is perfectly fine, even if the purpose is different.
It's not likely to come up but it's best to have all the documentation you would need for a fresh visa application (itinerary, proof of financial means, accommodation, return ticket…) with you when crossing the border so that if there is any doubt about your trip, you can easily show that you still fulfil all the requirements to visit the Schengen area.
If it was a single entry visa or you hadn't use it before, it's more delicate because it might appear as if you lied to get a business visa that you always intended to use for other purposes. But that's not an issue in your situation as you would already have used the visa for its main purpose. See also Should my first trip be to the country which issued my Schengen Visa? for all the details.