I have a player whose PC owns a bag of holding, and as part of a plan, has decided to hide in it for an ambush. According to the DMG, there is about 10 minutes of air inside the bag. The player know this, and has decided to keep the bag 'partially open', so there is a sort of vent to the outside world. For the sake of this question, if the bag is left with the flap open, does a living creature still have to worry about suffocation?
There are 10 minutes of air, regardless of whether the flap is open or not.
Bag of Holding (DMG p153-154)
... Breathing creatures inside the bag can survive up to a number of minutes equal to 10 divided by the number of creatures (minimum 1 minute), after which time they begin to suffocate. ...
Magic items do exactly what they say in their descriptions and nothing else. There is nothing mentioned about leaving a flap open. You should not assume that air can flow freely across the dimensional portal in the bag's opening. This is of course subject to your DM's ruling on the matter.
What you're really asking is if air flows freely across the dimensional portal that is the bag's mouth. Sounds like it's up to DM interpretation, but the rules as written seem to suggest that air does NOT flow freely.
Here are related questions that might help you (or your DM) decide: Do objects have to be placed in the bag by an intelligent agent (e.g. a person)? If I opened the bag and tossed it in a river, would the river water flow into the bag, or does the bag have to be "activated" for something to enter it? (and can something enter it in a continuous flow?) How about an acorn falling from a tree right into the open bag? Now... is there a difference between a solid object and air -- such as there being a certain threshold of mass/solidity to cross the portal unaided? If you're in the bag, can you see out? (Why or why not?)
I have always thought of a Bag of Holding as an activated device, and the portal does not transmit anything unless it is "willed to". Thus an inanimate object, gas, liquid, etc. would not pass without being assisted by a will as well as the physical act. Under water it would just get wet externally, unless you actively scooped the water in. I never accepted that it could be opened from inside.
Unless you left it open, or it was somehow otherwise physically opened before you ran out of air, you would die. I read that as 10 min plus your normal process of suffocation (health, stamina, etc.).