How do I run a closed beta test for my mobile application? Development will be finished in 3 weeks.

You need early adopters of your app - your friends or followers via social media. Otherwise I'd try to contact some testing company to provide some beta testing.

You're going about this wrong and a lot of the advice here is also wrong, by worrying about the consumer experience first. Stop what you're doing and get out and talk to restaurants.

Make sure that restaurants are willing to offer rewards, and get signed commitments from at least 10 restaurants that they will do so, based on the metrics that they define matters.

Next, use the restaurants' existing social media outlets to get their own fans signed-up into private beta as an "exclusive privilege" to their fans.

If you can't get restaurants to do those two things at this point, you have a high likelihood of failure should you pursue development of an app without this important piece of customer development complete.

Happy to talk to you in a quick call to provide more details about what I'm describing above.

When the waiter asks if they want dessert, he should also drop a card with URL/qr code to the app.
Depending on your flow, it will point to your landing page or directly to testflight (where they register to download the beta, you get their email also from here).

They key is to have real early adopters.
With this system, you:
- make it easy to discover the app
- give them a chatting/funny topic (will remember experience)
- have "true" adopters since they are at that restaurant

...might also suggest to snap a picture with a restaurant mascot for the picture they share related to their dining experience.

Signup 5-10 restaurants as your test group, give them freebies to come on board. Thereafter end users. Period!