A cancelled credit card does not mean bills are automatically void or the recurring subscription cancelled, but it's a scale whether it's worth it for the company to try to collect or not.

If your rent autobill fails, I think everyone realizes it doesn't mean you don't have to pay. If the credit card for your gym membership expires, I think most people would likewise fear (and has been known to happen) that your bill would eventually be sent to collections. A $5/mo subscription? Not worth the time of the company to try to collect. A Salesforce or Adobe subscription on a card? I certainly wouldn't gamble on that, and would try to properly cancel the subscription.