UK citizen married to an Irish citizen. How much time can I spend in the EU after Brexit?

Because your wife is an Irish citizen, you are the spouse of an EU citizen. You will therefore continue to have rights under the free movement directive even after most citizens of the UK cease having those rights. In particular, as long as you travel with your wife (or to join her), you are a beneficiary of the directive. So in fact, very little will change for you as far as traveling with your wife is concerned. This would not be true if your wife were not an Irish citizen.

Because you qualify for free movement, you may remain indefinitely in any EU country provided that you have sufficient resources (or your wife is working, studying, or looking for work, but these don't seem to apply to you). You may spend up to three months in each EU country without having to register. You may be required to register with the authorities to remain longer than three months. The details of this registration depend on the country in question -- some don't require it at all.

The 90/180 rule that applies to most short-term visitors in the Schengen area does not apply to you. This is explicit in the Schengen Borders Code, which excludes beneficiaries of the directive from the provisions applying to nationals of so-called "third countries."

Would I need a visa to do this after Brexit for each country we visit, or can I only stay for 90 days?

The UK is going to be on the visa-exempt list for the Schengen area (otherwise known as "Annex II"). Because of that, family members with a derivative right of free movement may only be required to have a visa if their country of nationality is an "Annex I" country. In other words, you will continue to be able to visit the Schengen area by yourself for short visits without a visa, and you will not need a visa in the Schengen area when you are there with your wife.

(If the relationship between the UK and the EU should become so strained that they introduce visa requirements, then you would also need a visa to travel with your wife, at least in theory, but family members crossing at the land border are generally admitted without such a visa provided they have satisfactory evidence of their family relationship. Furthermore, the possibility of such a severe deterioration between the UK and the EU seems very remote.)

Make sure that your wife enters with her EU (Irish) Passport, so that you (as a, visa free, non EU Citizen) are entering and staying as a spouse of an EU Citizen.

A long as both of you don't remain in one member state longer than 3 months, there should be no problem when leaving.

You will probably have more problems with any coronavirus specific restrictions, that at present cannot be forseen.

Assuming both are at present (2020) United Kingdom residents, proper health insurance will be required. The European Health Insurance Cards (EHIC) will still be valid for your EU Spouse, but not for you if you enter after the 1st of January 2021.

If you have a United Kindom driver's licence, an international driving permit (IDP) to drive in some countries may be needed.

If you are driving a United Kingdom registered car (Motorhome), you might also need a ‘green card’ or valid proof of insurance and a GB sticker.

Sources:

• Visit Europe from 1 January 2021 - GOV.UK
  • Healthcare for UK nationals visiting the EU - GOV.UK

The worst-case scenario is ordinary Brits will be able to stay for up to 90 days per 180-day period, upon justifying the purpose and funds for the trip if asked (I can imagine the French will be extremely lax about requiring documentation of British citizens).

As a family member of an EU national, however, you'll continue to be free to enter with no restrictions if accompanied by, or travelling to join, your wife. @phoog's answer covers it in detail.