Is there a word for "all but the first one"?

You could look in a thesaurus for an antonym of initial, but I just did and didn't find much. For what it's worth, in functional programming (e.g., Scala) the first element in a list (or any sequential collection type) is called the "head," and all subsequent elements together are called the "tail."

While it is apparently not used as much as non-initial, Non-first is an option that is used technically and can be general-purpose:

In the data, we find speakers using HT for producing non-first list items only. (Time and Emergence in Grammar: Dislocation..., by Doehler, Stefani and Horlacher)

or

Also, Nokia’s big payday is now a non-first quarter affair. (techcrunch article)

and

Therefore, due to the different location in a channel gain period, the sensing slots could be classified into two categories: first slot and non-first slot, and they are defined as (6). (EURASIP Journal on Wireless Comm...)

The first run was successful, while non-first results varied.

Edit including suggestions (Thanks, @user253751), just to keep keywords for the search engines:

"Non-first quarter" should be parsed as "non-{first quarter}" rather than "{non-first} quarter", making this construction the odd-ball in the three preceding examples. Another example is "Non-first order coupling". Non-x can be ambiguous at times; one wonders if the "non" (negation) is distributed or the "non-first". Anyhow, in the "order couplings" case I think it's fairly clear that the order is: "{{non {first order}} coupling}, or in another notation: Non-{first order} coupling "First order" has a specific meaning which "non-" is trying to negate. That is, check whether the coupling is first-order, and conclude that it's not - rather than checking which order the coupling is, and concluding that the order isn't 1.

In computer science, there is a kind of algorithm that works on lists, and makes exactly the distinction between the first and all other elements of a list.

head is the term for the first element.

tail is the term for the remaining list after removing the head.

rest could be an alternative to tail.

I'm not sure that works for your practical case, but it's related because the whole concept of lists is based on exactly the distinction you make.

The terms are used like this in the context of functional programming.
There are shell commands with the same names, working on lines of text files, which are related, but different.

As has been said in the comments, subsequent is the word.

The tests are successful only on the first run. On all subsequent runs, the tests fail.

In fact you could get away with:

The tests are successful on the first run. On all subsequent runs the tests fail.

Nobody could possibly be confused about what this means. There is no ambiguity.

All tests apart from the first one fail.

All tests other than the first fail.

All tests aside from the initial test fail.

"Repeat tests failed." works. Note that this is different from "repeated tests failed". The latter is similar to "multiple tests failed" in that it includes the first test, though it differs from "multiple" in that the tests are of identical kind.

Another possibility is "followup tests" but that also suggests tests of a different rather than the same kind.

I personally prefer "subsequent" over "noninitial" (a word I've never heard used though I reason it could be parsed correctly by a reader) but I also accept that "subsequent" doesn't stand on its own.

It doesn't mean that one has to use a full introductory sentence though, as the introducton could be via a single word in the same sentence as the subsequent by specifically declaring which test was the initial failure and leaving subsequent to pick up the remaining failures:

Second and subsequent tests fail

This approach also retains the flexibility of being able to vary the number of items that succeed, i.e. declaring the initial failure as attempt number 3, with "Third and subsequent.."

Once you've set the context by stating something about the first run, you can use "rest" to refer to all the remaining runs.

The tests are successful only on the first run. All the rest fail.