For meeting NEC code. Kinda yes, Kinda no. For the most part, if you don't make any changes there are no upgrade requirements. There are places in the code that says if a particular type of equipment is replaced or even repaired that part of the system must be brought up to a particular code. However, I have never seen anything in the code that requires upgrades to unrelated parts of the system.
Local building departments are a whole different story. They can require upgrades to unrelated items. Some building departments figure they can require anything they want when you ask for a permit. I once had to add a sewer cleanout at the street just to get a permit for a room extension that had nothing to do with the sewer. When I asked why they just said "Because it is their policy". (I knew better than to try to fight it..... they have all the power)
In the OP's case, I could easily see an inspector requiring the whole panel to be swapped out if anything at all is done to the panel. (and, IMHO, it would not be that unreasonable of a requirement)
When it comes to insurance, it is entirely up to the insurance company. They can refuse to ensure it or require the owner to pay more if it is not upgraded. Often, they just assume what is in a house based on when it was built and the owner does not even know they are paying more because of the assumptions. My son just got his fire insurance significantly reduced by adding an earthquake gas valve on his 1914 house. (Earthquake valves are required for new construction where he lives). He added the earthquake valve because he wanted to, but the guy he hired to do it told him to check with his insurance.... and sure enough, he got a reduction in his rates. (No one even asked about the gas when he first bought the house)