Are we blessed when people hurt us? Matthew 5:11 says,


  Blessed are ye when they shall revile you, and persecute you, and speak all that is evil against you, untruly, for my sake: (Douay-Rheims)


From the sound of it, this verse has mostly to do with what people say or think of us. But what about when people beat us? or steal from us? or poison us? Does the same blessing apply in those cases as well? Would they also be examples of persecution?

Are there other verses from the Sermon on the Mount, or the Gospels, or the NT, that deal with the theme of injury and the blessing that comes with it?
You are roughly correct, but with a condition I'll highlight later. 

When you are beaten, stolen, poisoned, and the world reviles you, you have less and less that is in between you and God. It's very evident that the more gifts, accolades, status, riches one has of the world, they only serve to distance that person from God since it reinforces the notion that their own way, and not Jesus is what can provide their own Justification. Thus, the logical statement is If (riches of the world) then (generally further from God), the logical contrapositive states if (closer to God) then (no riches). The better way to say this is if you hold anything above God as closer to you, such as money, then you are further away from God. Thus, being hurt, beaten, stolen from, etc.. doesn't necessarily make you blessed per se, but rather, puts you in a position so that there is less between you and God. In a sense, the blessing is that you now have the position to be closer to God because you have less things of the world (ego, status, Money, health) to hold close to your heart. 

Consider the rich ruler in Luke 18:18-29. Jesus tells him "Sell all that you own and distribute the money to the poor, and you will have treasure in heaven; then come, follow me." not because he's giving him a set of orders and 10-step-easy-guide-to-heaven, but rather, he knew what was in his heart - Money. And that the rich ruler held money closer to his heart than the Lord. So Jesus simply revealed to him what he truly valued in his heart.  If you are hurt, you are blessed in a sense because you certainly don't have self-pride getting in the way when you're on your back and out. Also, the Holy Spirit is closer when we are weak Romans 8:26.