Do exit immigration officers care if you’re heading to a country you are obviously not allowed to visit?

Do immigration officers care when leaving to a country you are obviously not allowed to visit?

Typically no, but that's probably not universal and they are likely some exceptions to the rule.

Any exception would be likely at a land border since enforcement at ports or airports are handled by the airlines or cruise ship lines and most governments put hefty fines in place to make sure they do a thorough (but not perfect) job at it

Consider a land border from A to B. Exceptions could be triggered by cases where any problem entering B would also affect the country A. Let's say you have a single entry visa for A and no Visa for B. At exit immigration, the officer country A would check that you were legally in the country have not overstayed AND they would invalidate your single entry Visa.

If you now move on to country B, they won't let you in since you don't have a Visa. Typically they would just sent you back to A, but since your single-entry Visa has been invalidated you can't go back there either. So you are stuck between A and B and someone has to deal with you.

In many cases, the exiting country has a process for this, where you would get an emergency transit Visa or some such, but that's a lot of extra hassle and paperwork and so it's a lot easier for country A if the exit immigration officer checks your credentials for entry into B BEFORE they invalidate your single entry Visa.

Another example: Exit immigration in Hongkong does a fairly thorough check of your papers at any land border to mainland China. I'm pretty sure they would refuse you exit if there is a problem with your entry credentials into China. Granted, I have never tried, and I'll make darn sure I never will!

Overall there are 186 options for country "A" and for country "B", 200 or so possibilities for your citizenship plus a many different visa options and all of these are governed by a different set of laws and rules. These are tens of millions of combinations and making any general statements is always likely to be incomplete

Immigration of a country would not know all other countries rules, for potentially everyone from any other country - a huge complexity, and ever changing. They do not care.

For flying:
If you cannot travel to a specific country with your passport/visa/documentation (without getting a visa/waiver/etc. first), the airline is responsible to not let you board. This is part of their agreement to even be allowed to fly people into the target country. Airlines typically use the TIMATIC database to verify your eligibility to tarvel to a country. [note that it is your responsibility to have all necessary documents - the airline will simple decline you to board].
In practice, you would never make it to immigration, as you are not allowed to get a boarding pass or enter the secure area.

For land borders:
As you are going in one specific country, immigration probably knows the rules, and might warn you - but it is not their responsibility. There are known cases of people stuck between borders, not being allowed in either country; although typically the country you came from would let you back in, reinstating your previous situation (and put you in jail, if it was illegal)