I have read through Niantic's support regarding the new feature which allows you to send and receive gifts. Based on the support link, you are limited to sending a single gift per friend each day.
You can send each of your friends up to one Gift per day
You are also limited to how many gifts you can hold in your inventory before acquiring another from a PokéStops.
You can hold a maximum of 10 Gifts in your Bag at any given time
However, the support does not mention a limit on how many gifts you are able to open per day.
Is there a limit to how many gifts I can open per day? i.e. If I have 15 friends who all send me a gift in a single day, am I able to open all 15?