TL;DR: maintain your home and centre of life in Germany, and only work abroad temporarily. This should be sufficient to maintain tax residency in Germany, but won't exempt you from all Irish taxes.
As a general rule of thumb, tax residency immediately follows your actual residency. If you move to a different country, you're covered by that tax regime. However, it is not entirely clear in your scenario that you would become resident in Ireland. There are also interesting collisions when you move during a year, when you work and live in different countries, and when certain tax obligations do not expire immediately after moving out of a country (in order to combat tax evasion).
In a German context there are two kinds of tax residency: unbeschränkt einkommensteuerpflichtig (~ unlimited tax obligation) which would normally be understood as “tax residency”, and beschränkt einkommensteuerpflichtig (~ limited tax obligation).
Beschränkte Einkommensteuerpflicht: This applies if you have certain kinds of income from Germany but you do not have your habitual abode in Germany and have no German residence (as in: place to live). In most cases, this is overridden/clarified by tax treaties (which will be discussed below).
You might have relevant German income, but the interesting question is whether you would have a German residence. You indicate that you intend to continue renting a German flat. If this is clearly your main place to live, despite temporarily living in Ireland, then this case might not apply.
Unbeschränkte Einkommensteuerpflicht: This is the normal tax status for people who live in Germany. The important question is whether you have your habitual abode or residence in Germany. As discussed in the previous paragraph, this is not entirely clear.
A person who would otherwise have only a Beschränkte Einkommensteuerpflicht can opt in to Unbeschränkt status, but only if the non-German income is insignificant. This would not be the case for you when working in Ireland. Unbeschränkt status can also be chosen when temporarily living apart from a spouse who has Unbeschränkt status in Germany, so that the married couple can file taxes together.
Fortunately, the Irish–German tax treaty provides a concrete test to determine where you would be resident:
Where by reason of the provisions of paragraph 1 an individual is a resident of both Contracting States, then his status shall be determined as follows:
a) he shall be deemed to be a resident only of the Contracting State in which he has a permanent home available to him; if he has a permanent home available to him in both Contracting States, he shall be deemed to be a resident only of the Contracting State with which his personal and economic relations are closer (centre of vital interests);
b) if the Contracting State in which he has his centre of vital interests cannot be determined, or if he has not a permanent home available to him in either Contracting State, he shall be deemed to be a resident only of the Contracting State in which he has an habitual abode;
c) if he has an habitual abode in both Contracting States or in neither of them, he shall be deemed to be a resident only of the Contracting State of which he is a national;
d) if he is a national of both Contracting States or of neither of them, the competent authorities of the Contracting States shall settle the question by mutual agreement.
(from Art 4(2) of the Irish–German tax agreement from 2011-03-30)
Note that the treaty gives your permanent home and vital interests priority over your habitual abode (gewöhnlicher Aufenthalt) for the purpose of determining residency. This is good for you, because your habitual abode would definitely follow to Ireland if you're staying there for more than six consecutive months (short interruptions don't matter).
By continuing to rent your German flat, you would likely have a permanent home available to you in Germany. However, access to the family home in Ireland might also be a permanent home available to you (it doesn't matter if you're renting it or not). In that case, your centre of vital interests would be relevant. That you run a business in Germany and that you intend to return after a year or so is a good indication that you are centred around Germany. But note that your centre of personal relationships also matters – that your family home is in Ireland could weigh in Ireland's favour. You seem to not be German, so it might take a bit more effort to demonstrate that your life is centred around Germany.
An 183 day rule exists, but it's not about tax residency but about reducing bureaucracy relating to income from employment. Such income is almost always taxable in the country where it was earned. So your income from teaching in Ireland will be taxed in Ireland from day 1. Per the tax treaty, this income would only be exempt from Irish taxes if you were in Ireland for less than 183 days per tax year and the employer didn't have a permanent establishment in Ireland. If the teaching were a self-employed position, the question would be solely where you have a permanent establishment such as an office.
If you're teaching in Ireland at the invitation of the Irish state or an university, college, school, museum or other cultural institution or under an official cultural exchange program, then Art 19 of the tax treaty would largely exempt you from the Irish tax regime. But there is a strict purpose limitation for your visit (teaching, lecturing, and researching) and it is limited to at most two years. This would be an elegant solution to you since it side-steps most of these tax complications, but getting an invitation from a suitable institution might prove difficult. And this exemption only covers remuneration for the covered activity, not e.g. income from entertainment or musical activities.
A “90 days in any four-year period” rule seems unsubstantiated and would contradict other rules.
What should you do with this information?
- Your situation is quite complex. I hope I could explain some background, but ultimately you have to convince the tax authorities of your interpretation.
- You will greatly benefit from an Irish tax professional who can help you convince the Irish authorities that you're not tax-resident there. You will still have to pay Irish taxes for your Irish income (unless you're an invited lecturer), but this wouldn't affect your unbeschränkte Steuerpflicht in Germany. Different parts of your income can be taxed in different places.
- In line with the advice from your Irish tax professional, you should take steps to clarify that you're resident in Germany. For example, I would expect that you visit your German flat multiple times per year in order to manage your affairs.
- If you can get an invitation to teach at a suitable institution such as a college, that could greatly simplify your tax filings, though it has nothing to do with tax residency per se.