Is there any country in the world where you can enter by scanning your fingerprints/iris, without showing any forms of ID?

If you have a Nexus card, this is how you enter the US from Canada, or Canada from anywhere. You go to a machine, push a few buttons, look into the lens so your iris can be recognized, and are given a little receipt which you can show people as you leave the area. You are supposed to carry your Nexus card (and it saves you from having to carry your passport) but in the normal course of events you will not need to show either one to a human or put either one into a machine. It's all done by your iris.

I have done this more times than I can count. I have been doing so for ten years, and it is only this year that I got an enhanced passport with a chip in it. My passport doesn't include iris information - I know because I didn't give them any.

The Privium system at Amsterdam airport does not require you to insert your passport in the machine or to show it to anybody. You do need a special Privium card, which contains the biometrics data, and are still supposed to have your passport with you, obviously. Enrolment in this system is voluntary (and starts at €121 per year). The regular automated passport control at Schiphol works differently and do rely on the passport's optical machine readable zone.

There are advantages to the use of passports for automated border checks: You don't need to install and secure widespread access to a sensitive database and you can support passports from other countries (for which you don't have access to any central database). In fact, you don't need any central biometrics database at all, which has clear security and privacy benefits.

Generally speaking, note that matching a person to a known set of biometrics (whether you read them from a chip or look them up in a database based on name and date of birth or a special identification badge) is a completely different problem than a wide search through a large biometrics database. The latter is significantly slower and brings up many false positives with the data available now (picture of the face and fingerprints). It can be used for investigation purposes, for things like asylum applications and in a few other contexts but it would not be practical for automated border checks.

Importantly, not all countries retain the data that's on the passport. And when they do, they do not always keep all the data, what they keep might not be centralised, or it might not be available for automated processing. Apart from India, it seems that no country systematically collects iris data either. Considering all this, it seems that your question is based on a rather optimistic (or pessimistic, depending on how you feel about all this) view of the state of biometric data collection and processing.

Earlier this year Australia announced plans for a contactless passenger identification system:

Australia is planning to adopt a new contactless passenger identification system that would eliminate the need for passport scanners, paper landing cards and manned immigration desks, the Australian Department of Immigration and Border Protection has announced.

However:

It is unclear exactly how the new contactless system would work. In fact, even the Australian Department of Immigration and Border Protection seems unsure how its new system will actually function.