Insert big YMMV tags all over this, but here's my perspective.
Yes and no. It won't be forgotten, but it'll probably be quickly forgiven - particularly if you appear to be aware that you made a mistake and why it matters, and don't make a habit of it. Once it's fixed, it's water under the bridge. Interpersonal conflicts, not so much. They'll be moved on from, considered no longer relevant, and they probably will never be brought up again even in the heat of an argument, but they will not be forgotten and they may well smoulder very quietly for years and years. Hurt feelings rarely get allowed any airspace, but they tend to just retreat to a deeper, hidden level rather than actually vanishing.2. once the issue is solved, does the INTP retain the mistake in his head? from my prior experience with the boss, I think yes, but sometimes he has selective memory.
Haha! We do make mistakes, it's just that competence is so hard-wired into our identities and self-worth that when we slip up we do everything possible to conceal, fix and excuse it. One of the worst things you can probably do to an INTP is make them feel incompetent at something they care about being good at. Conversely, appreciating their contribution - particularly their more unique and off-the-wall contributions and analyses - is a good way of making an INTP feel loved. (Lots of people are inclined to respond to an INTP's abstract analyses and unusual ideas with a expression, which is sad.)3. how do you get approval from an INTP? I feel that he really likes other T types. Those trusted in my office are INTPs and an ENTJ. I try masquerading as a T type but I feel like T types never make mistakes?!!
INTPs who haven't worked on their Fe can be hard work for Feeling types, as half the time we're operating in a kind of theoretical space at at least one remove from the world, so it doesn't even occur to us that what we've just said might be hurtful. This is important: when we're abrasive in casual conversation, as opposed to in an argument, we usually don't mean to be nasty. (We can be extravagantly nasty when we want to be, of course...) A lot of the time, we've just got carried away with our wordplay and crossed the line from witty to biting without realising. Often the best way of responding then is to simply say "I realise you probably didn't mean to be sharp just then, but that was actually quite hurtful/a bit too close to the bone/something I'd rather you didn't joke about if you don't mind." Chances are they'll be mortified and apologise, and they may well remember and carry that mistake for years and years to come, long after everyone else has forgotten it. This particularly works if you frame it as a simple statement and acknowledge that it probably wasn't intended, rather than getting angry or bursting into tears, which can make INTPs flail, panic and go on the defensive. And then accept their apology gracefully and move on. It's our least favourite function, but we do extrovert Feeling, and on the whole we'd actually rather everyone got on well with each other. There's so much less messy drama that way!
INTPs tend to value and respect originality, both of concepts and approach, most especially when they're supported by reasoning. Doesn't necessarily have to be empirical facts, but there needs to be a thought process there. They don't tend to have much time for "I think x because it just *feels* like x", and may well dismiss your idea out of hand (and peg you as fluffy-brained) for the crime of justifying it with feelings. "It just feels that way" can look like a flashing sign saying "You can't analyse this", which is frustrating "I think x but I'm having difficulty working out exactly why" frames what's essentially the same statement in a way that invites T to pick it apart, which is something INTPs are inclined to latch onto. Similarly, you can get INTPs to treat feeling-based issues and ideas seriously, if you can present it in terms of "People tend to do this, which could have this effect on this project, so that's a factor that might need accounting for in practice." Being ready to pick up their analysis or apparently-zany idea and play with it til you see where it might go is a good way of endearing yourself to an INTP co-worker. Conversely, knee-jerk reactions without turning the idea over properly, and slavish adherence to tradition and the "normal way of doing things" are likely to frustrate us.
INTPs do not like being micro-managed, hate being nagged, and they don't like micro-managing much either in my (admittedly limited) experience. They expect people to think for themselves. On the other hand, they tend not to be great at keeping track of things, and people who will do that for them without needing the INTP to chase them are really valuable. This is going to vary hugely, but as a rule of thumb I'd just keep them abreast of major developments that are actually going to have an impact on the INTP. This is harder work for you, but if you can learn to second-guess what they think is important and keep them up to date on that, and have answers ready when they inevitably forget and ask you about it, you'll be worth your weight in gold. Irrelevant information is annoying and makes us think that you can't prioritise, but at the same time, we don't like having to chase the important bits. (INTPs are poster children for why having a good PA is half the battle. )4. should I regularly go to the INTP and give him updates with projects? or is that annoying?
No. Conspicuous tiptoeing makes us think that you're making a huge deal out of the issue, which is Interpersonal Drama and makes us uncomfortable. Giving us a little space immediately afterwards is fine, but really the very best thing you can do for an INTP after cocking up is do something well. Bonus points if it's something we'd forgotten about! We know that people make mistakes; all you need to do is fix the issue if you can, and reassert your competence. On the other hand, if it's the INTP who's cocked up, I highly recommend a) giving them space b) being normal and friendly when you see them. We hate getting it wrong, and while asking us to cope with lots of people when we're beating ourselves up for Failing is a big ask, evidence that everybody doesn't now despise us is balm. The sooner everyone's acting normally again, the sooner we can draw a line under the nasty messy feelings and move on.5. [I know the answer to this but...] should one leave you alone after a little mistake or is tiptoeing around after said mistake ridiculous?