Generally, no. If you have 3 birthday reminders and each of them has a coupon inside it, the subsequent ones won't be sent anymore after a coupon in a prior reminder has been redeemed. This is set to default for scarcity. But just to make sure or if you need to keep the reminders sending, there is an option that you can disable in the campaign properties so the messaging will not stop after a coupon in one of the reminders has been redeemed. 

First, go to your campaigns list and find the campaign that you want to check out. Once you've picked one, click the Actions menu button to open it up then click Properties

Once inside the Properties, open up the Message Settings tab. Now you'll see an option called Stop messages after a redemption?

If that is set to Yes, then the messaging will stop after a coupon has been redeemed.