Who keeps the bitcoins? If he didn't left to anyone his private key, will the bitcoins be lost forever and ever?
Yes, those coins would be lost. It would be as if you buried a chest of gold out in the middle of nowhere and died without telling anyone about it.
Though it is possible (though mindbendingly unlikely. Even if bitcoin keeps existing until the heat death of the universe, it will almost certainly never happen.) for someone to randomly recreate the keypair for those coins and thus take possession of them.
There is a solution for that in the market right now. There are Digital Legacy platforms that keep your Digital Data safe while you are alive and transfer them to your assigned beneficiaries when you are gone. What I mean by Digital Data is all your information and belongings you have on the web, such as - accounts, passwords, docs, pics, contact list, insurances as well as your online currencies (Bitcoins and other type of coins). I am currently using a platform called PlannedDeparture. It allows me to store all the information I want and assign beneficiaries with instructions with what I want them to do with that information or data. For example to close down my facebook account or to claim the money from my online banking and my Bitcoins too. The platform check your activity on social media (you choose which one). If you do not appear to be active they will send an email to verifiers that you have again assigned, to verify your current situation. Once they have confirmed your situation, They will transfer all the data to your beneficiaries with instructions. It is a very user-friendly platform that allows you to store many information under many categories. Check it out at www.planneddeparture.com if you dont like it there are others that may suit your preferences better.