A cloud-based service included in Microsoft 365, delivering scalable messaging and collaboration features with simplified management and automatic updates.
In my experience as soon as you add a custom domain to 365, Microsoft will start routing some MICROSOFT domain email such as support request responses internally to your tenant, and there is no way to then route them from your tenant to on-prem. You may want to use your onmicrosoft.com account or another external email domain account if you need to contact support in the interim before you start using the custom domain.
Also if you have another custom domain already in use in Microsoft 365 in that same tenant, you will need to change 365 to relay for your unconfigured local domain. In the 365 Exchange Admin > Mail flow > Accepted domains > select your in-use hosted domain and set it to Internal Relay.
As stated by Jame Xu-MSFT, other Microsoft tenants will honor your published DNS records and route correctly. It will mainly be an issue among some of Microsoft's servers - not all of them - for example licensing emails will get to you, but not support requests.
It may also become an issue setting up a new remote user in Outlook since AutoDiscover will check for a 365 tenant first. The workaround is to set HKEY_CURRENT_USER\Software\Microsoft\Office\16.0\Outlook\AutoDiscover\ExcludeExplicitO365Endpoint to 1. Just make a note to remove that once you do start using the local email domain in 365.