If Updating Product Handle in Shopify Breaks a product URL, you most likely need to re-deploy your site on Vercel, Netflify, or what ever CDN/hosting you are using.
It is not always the case that updating a product in Shopify will break a URL, but depending on how much of your site is pre-rendered this may be an issue.
What is Pre-Rendering?
Pre-rendering occurs when you deploy your site. If a page is set to be pre-rendered, then all of the content is gather during deployment, and arranged in its final form. This final form is then saved (as an HTML page), and is shown to customers who visit your site. Updates on Shopify, or any other CMS are not reflected on pre-rendered pages because the final page was built during deployment.
In order to reflect these changes you have to trigger a redeployment, which would force the page to reach out to Shopify, or another CMS in order to gather the required/updated content to build the page.
Conversely, dynamic pages make a request to the server every time they are called on. Because they are always reaching out to the server (for example Shopify), they are always getting the latest data.