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

If you sign up for CLEAR, you are allowed to fly domestically within the US without showing any form of ID to TSA agents. Instead, the CLEAR machine does an iris scan (which takes around 5 seconds) to confirm your identity. In theory you’re still supposed to carry a form of ID with you, however I was able to take multiple flights around the US without ever taking out my ID.

This is a tangential answer because it only applies to domestic flights within the US.

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.

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.

As luck may have it, I’ve found such a system after posting my question.

Iris Recognition Immigration System (IRIS) was an initiative, launched in 2004 to provide automated clearance through UK immigration for certain frequent travellers. It functioned in “one-to-all” identification mode, searching a large database of some million enrolled frequent travellers to see if anyone matched the presenting iris. The passenger was not asked to assert any identity, e.g. by presenting a Passport or ID card which would then require only a “one-to-one” verification test against that single enrolled identity. Thus IRIS exploited the great robustness of iris recognition against making False Matches, since (unlike weak biometrics such as face recognition) it could survive the vast number of opportunities to make False Matches when searching a large database, instead of only needing to test just a single asserted identity.

Unfortunately it was decommissioned by the UK back in 2013:

It was decommissioned in September 2013 in favour of e-passport entry using biometric data stored on the e-passport chip, which requires only a single asserted identity to be tested and thus could function using weaker biometrics in the simple “one-to-one” verification mode.

The biggest problem seems to have been the need to collect iris scans from all applicants:

But the main factor behind the decision to decommission IRIS was the cost of staffing the enrolment offices in airport terminals, whereas face recognition only required submitting a Passport-type photograph acquired in a photo-booth without the need for personal presentation to airport enrolment staff.

We might see a comeback of the technology once iris scans are routinely collected from all citizens when they are issued a biometric passport. Until then fingerprint and facial recognition technologies are too slow and have too many false positives in order to be used for comparisons against a database of all citizens.

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.