Is there a limit to how many gifts I can open per day?

There is a limit of opening 20 gifts per day. You can stack them to open up a next day in case you get too many.

Source:

Be aware that Niantic has set a limitation to this feature and you can’t open as many gifts as you’d like: you can ONLY open up 20 gifts per day.

You'll get the following error message if you try to go over the limit:

error screenshot