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Hard-Brexiteers would like the UK to have a "clean break" with the EU, and in particular they don't want any part of the UK to stay even temporarily in the customs union (the so-called backstop).

This "clean break" would normally require a hard border between Northern Ireland and the Republic of Ireland, but a hard border would breach the Good Friday Agreement. It seems that the options "the UK leaves the customs union" and "the Good Friday Agreement is upheld" are mutually exclusive.

As far as I'm aware, the only idea that some hard-Brexiteers have proposed to solve this contradiction is a soft border using "seamless technology". Even assuming that such a technological option is feasible for the trade of goods, wouldn't that leave a backdoor wide open for illegal immigration from the EU to the UK? If yes, are hard-Brexiteers ok with that? Alternatively, are there any hard-Brexiteers who support a hard border?

To summarize, what do hard-Brexiteers see as the ideal outcome for the Irish border in the long term?

They tend to think it's somebody else's problem (Ireland's and/or the DUP's). Unless you are part of the DUP of course. See how Rees-Mogg has been punting the problem along the lines of: I agree with whatever the DUP agrees (or at least doesn't oppose) on Northern Ireland. And at the same time he says that in the case of no-deal Ireland would not dare to impose a hard border. Which is true to some extent.

The DUP has given a number of somewhat contradictory statements on this, over time. From the somewhat famous denial that a hard border ever existed to the more recent position(s) that they would prioritize staying the EU over splitting Northern Ireland from the UK.

I'm a Remainer, but have been accused of being a Hard Brexiter on some stances in this debate, so I'll take a stab. The WTO, the Repubic of Ireland, and the EU have all recently made declarations and supporting statements that they have no intention of putting up border checks on any border between the Northern Ireland and the Republic of Ireland. They will instead impose checks at warehouses and other centers of of commerce. The UK is adamant that they will not impose any border check on any borders between ROI and NI (already been there and done that) and currently, through the HMRC, conduct such checks (for immigration and contraband) in Northern Ireland. So it seems that this issue is, perhap ironically, sorting itself out whether there is a 'hard' Brexit or not. The means by which such controls can be done are long established by many nations around the world (a simple Google search with near infinite examples, though if enough people insist, I'll post all sorts of links to customs seizures and immigration raids away from any borders). It will not be perfect. Nobody has a perfect system that I'm aware of. But it will suffice to control immigration and movement of goods to some extent and, most importantly, prevent any return of the Troubles.

The brexiteers don't really want anything regarding the Irish border. It's just a problem preventing them getting the hard brexit that they want, and since they don't have a real solution for it they just want to pretend it's not really a problem.

That's all it is, an annoying roadblock for them.

I think it's quite telling that the top answer here begins while speaking in the third person. In the long running Stack Overflow tradition, allow me to tell you that there is something wrong with your question.

First: there are no hard brexiters. There are only people committed to securing Brexit at any cost necessary and see through the tactics adopted by Brussels to punish the UK for seeking independence by denying (and not "not allowing", no, specifically denying, because a bilateral trade deal is a minimum requirement of any civilized nation, never mind next door neighbours, it's not the 1600s) the UK a trade deal and access to the common market, which, according to basic economic theory, punishes the EU more than it does the UK (as the UK is a net importer, not exporter, thus why Germany has sounded the alarm on all fronts in the last year, and this is before Brexit even took place (https://www.dw.com/en/german-economic-growth-to-drop-to-1-percent/a-47292722)); but Brussels cares not, as Brussels and the EU bureaucracy are very separate from Europe, never mind member nations of the European Union which are completely removed from the conversation.

Second, your question concerning the Irish border question is irrelevant. Most Brexiters do not think about it, as, due to eponymy, their goal is to secure Brexit, and not to alter the status quo in any other way.

TL;DR Your question is misplaced, and Brexiters A) do not care about the Irish border and B) want the current status quo to remain as is, which in terms for the Irish border will either mean a hard border, or a one-nation exception to be granted for Ireland, which will in turn pave the way for a re-integration of Ireland back into the United Kingdom.