Facebook SDK iOS invite user without FBWebDialogs

There is option of frictionless requests.
On Facebook developer site they mentioned in section of Invites and Requests

We touched on a scenario where users exchange requests back and forth. If this scenario is typical in your game, it can be a bad user experience to force them through the request dialog every time they want to send a request. The solution for this is frictionless requests.

Frictionless requests let users send requests to friends from an app without having to click on a pop-up confirmation dialog. When sending a request to a friend, a user can authorize the app to send subsequent requests to the same friend without another dialog. This streamlines the process of sharing with friends.

For more reference see Facebook Invites and Requests

You are asking two questions here:

How to invite Facebook friends without web dialogs?

Unfortunately, the web dialogs you are referring to are the best way to send invites to friends. There was a time when we had to resort to ugly hacks just to get this functionality in applications. Facebook added this functionality to iOS SDK after developers created bug reports and were literally behind them.

Get a list of friends who are not using the app and invite them

Foursquare IMO is not using the requests API. What they do is, whenever somebody links their Facebook account to their Foursquare account they make a note of the users friends who are using/not using the app. This is possible if the user grants the app permissions.

Then in the invite screen, they simply build a UITableView with list of friends who they think are not using the app. When you tap Invite they will just send an email invitation and not the Facebook request you are referring to.

I think FourSquare syncing the user's friend list(friend ids) to its own server.. Then after they are checking it to create the custom interface like the screen shot you shared.