Presuming that OP is a US passport holder and that domestic refers to the US.
It goes without saying that you do not need a passport to travel on domestic flights in the US, although a passport is valid ID.
For international, one of the reasons why most carriers do a quick second document check at the boarding gate is for exactly the circumstances you describe. Even if your documents were thoroughly checked at the counter or elsewhere, checking that you have your passport as you board would generally lead to being denied boarding if you have lost it between check in and the gate. It is then up to you to get it replaced and to rebook for some other time.
If you are talking about a US passport, it is possible to replace a passport (no emergency passports - those are to get you back home if you lose yours while overseas) in a few hours at a Passport Agency (there are 25 in the US) and those "few hours" are during regular business hours. Thus, if the next day is a business day, good chance you only lose a day. Depending on your fare rules, reissuing a cancelled ticket may be costly (or impossible).
If this happens overseas, the US cannot deny entry to its citizens, but the lack of a passport likely means you won't make it to the US to be admitted. Thus, US Consulates overseas are able to issue passport replacing documents (emergency passports). These are typically good for just long enough to get you back to the US and can't be used to continue your travel elsewhere.