I suspect that when our generation of Bitcoin holders dies, a lot of us will not have left information for our heirs to access our wallets.
Why not? It's not like you can take it with you.
If so, then most of the Bitcoin supply will eventually become inactive as most wallets containing Bitcoin will belong to people who are deceased.
Unlikely, but theoretically not impossible.
As mining will eventually finish as well, there will be no way to generate new bitcoins. The TOTAL supply and market cap will look the same on paper but, in reality, the ACTIVE supply and market cap will be a lot less due to these inactive wallets. Eventually, as the active supply falls, the price should rise but it will not reflect the total supply just the active supply.
Correct. This is called "deflation" and is why bitcoin is considered to be a deflationary currency in the long run. It was intentionally designed this way, and anyone that chooses to use it is voluntarily choosing to participate in a deflationary currency whether they realize it or not.
After all Bitcoins are generated, do you agree that the ACTIVE supply will eventually fall
Slowly. Yes. This is intended behavior and is how bitcoin was intentionally designed to work.
and is there a way to stop it?
Not without a hard fork and agreement from the vast majority of all participants (miners, nodes, merchants, users, exchanges, speculators, etc). Getting that sort of agreement on changing the very nature of bitcoin, a nature that most of them desire in the first place, is effectively impossible. It isn't going to happen. Why would you want to stop it anyhow? This is one of the great features of bitcoin.
In the long run, if 50% of Bitcoin’s total supply becomes inactive will people still want to use it?
Absolutely! Why wouldn't you? You get to have the exact same number of satoshis, but they are worth twice as much money. Isn't this a good thing?
Is there a way to measure active supply?
Not precisely. You could look at "bitcoin days destroyed" or "total bitcoins that haven't moved recently", but that wouldn't distinguish between bitcoins that are inactive due to being "permanently lost" and bitcoins that are inactive due to being "stored for later use".
Can crypto-currencies be programmed to automatically burn all coins in wallets that have been inactive for more than 100 years?
They can, but I wouldn't use one like that. It won't happen with bitcoin.