No, Tinder does not post to your Facebook at all.
In fact nothing happens on your Facebook profile at all when you connect with Tinder.
The dating app does require you to sign up with Facebook though, and you'll only be allowed to use images on the app that have been used as your Facebook profile photo.
Why does Tinder use my Facebook photos?
I think this helps prevent people from "cheating" by using photos that aren't actually them. It's a problem many dating sites have apparently...
What does Tinder do with my Facebook details?
Tinder also looks at the things you've liked on Facebook (such as books, movies, hobbies etc...) and if someone that views your Tinder profile shares those interests, then they'll appear to them on your profile.
The idea is to encourage people with shared interests to connect.
How secure is Tinder?
Pretty safe overall. The app doesn't do anything silly like share your Facebook profile or allow people to see information that would enable them to contact you outside the Tinder app environment.
There have been some technical flaws with the app in the past that meant some data was accessible to hackers, but they seem to have shut that problem down pretty quickly - and it's nothing that any app would be susceptible to.
Overall, I'd say Tinder is a very safe app to use, and you have no reason to worry about it posting to your Facebook profile or sharing anything about you that you hadn't specified.